TOWER, SMITH, TURTON,
AND
COPE

[Pg i]
[Pg ii]
[Pg iii]

BY

WILLIS E. TOWER, M. SCI. (Univ. of Illinois)
HEAD OF THE DEPARTMENT OF PHYSICS, ENGLEWOOD
HIGH SCHOOL, CHICAGO

CHARLES H. SMITH, M. E. (Cornell)
HEAD OF THE DEPARTMENT OF PHYSICS AND ASSISTANT
PRINCIPAL, HYDE PARK SCHOOL, CHICAGO

CHARLES M. TURTON, A. M. (Syracuse)
HEAD OF THE DEPARTMENT OF PHYSICS, BOWEN
HIGH SCHOOL, CHICAGO

IN COLLABORATION WITH

THOMAS D. COPE, Ph.D. (Pennsylvania)
ASSISTANT PROFESSOR OF PHYSICS, UNIVERSITY
OF PENNSYLVANIA

BASED UPON
PRINCIPLES OF PHYSICS
BY
TOWER, SMITH and TURTON

WITH 7 PLATES AND 448 OTHER ILLUSTRATIONS

PHILADELPHIA
P. BLAKISTON'S SON & CO.
1012 WALNUT STREET

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Copyright, 1920, by P. Blakiston's Son & Co.

In the preparation of this text, the pupil, his experience, needs, and interests have been constantly kept in mind. The order of topics, illustrations, and problems have been selected with the purpose of leading the pupil into a clear understanding of the physical phenomena continually taking place about him.

The recommendations and conclusions reached by the "New Movement in the Teaching of Physics" have been incorporated into the book as a whole. These conclusions indicate that the most efficient teaching in physics involves a departure from the quantitative, mathematical methods of presentation that were in general use a dozen or more years ago, toward a method better adapted to the capabilities, interests, and requirements of the young people in our physics classes.

The older methods are effective with a portion of the student body which has the greater mathematical ability and training, but they discourage a large majority of the pupils who are not gifted or prepared for severe mathematical analysis. For this reason, many of the more difficult mathematical demonstrations often given in physics texts are omitted. Most of the problems involve only the units employed in practical every-day measurements.

The portions of Mechanics that are ordinarily so difficult for the average pupil are not taken up until he has covered considerable ground with which he is more or less familiar and not until he has become somewhat accustomed[Pg vi] to the methods of study and the technical terms of the subject.

The pupil comes to the study of physics with a great number of experiences and impressions of physical phenomena continually occurring about him. In recognition of this fact, it has been thought best to consider first the explanation of common things well known to all pupils, such as the diffusion of gases, evaporation of liquids, expansion of bodies when heated, and capillary action. Since the molecular theory of matter is now supported by so many conclusive evidences, we have not hesitated to make free use of it in the early chapters. The applications of this theory are extremely helpful in explaining every-day phenomena. Our experience shows that beginners in physics understand and apply this theory without difficulty.

The illustrations and drawings have been selected from a pedagogical rather than a spectacular point of view. Practically all of them are new. The problems and exercises have been selected for the distinct purpose of illustrating the principles taught in the text and for their practical applications.

Many direct applications to common every-day experiences are given in order to connect the subject matter with the home environment and daily observation of physical phenomena. Some phenomena are mentioned without detailed explanation as it is felt that the presentation of these subjects in this manner is better for this grade of student than a complete analysis.

Some of the special features of the text may be briefly summarized as follows:

(A) Simplicity of presentation is emphasized. The methods of attack, the illustrations and examples employed in developing the subjects are particularly adapted to beginners in physics.

[Pg vii]

(B) The text is divided into some seventy-seven sections, each containing material enough for one recitation.

(C) Each of these sections is summarized by a list of important topics which point out to the pupil the principles and subject matter requiring most careful attention. The lists of important topics are also of assistance to the teacher in assigning recitations.

(D) The problems and practical exercises emphasize physical principles as distinguished from mathematical training. A list of exercises is placed at the end of the several sections. They are in sufficient number to permit testing at many points and of a choice of problems by teachers.

The authors wish to express their appreciation for suggestions and helpful criticisms to many who have read the text in manuscript or proof. Especially to Professor A. P. Carman of the University of Illinois and his associate, Professor F. R. Watson, who have gone carefully over the whole text; and to Mr. Chas. M. Brunson, Scott High School, Toledo, Ohio, Mr. Frank E. Goodell, North High School, Des Moines, Iowa, and to Mr. Walter R. Ahrens, Englewood High School, Chicago, for assistance in reading the proofs. Also to Mr. W. H. Collins, Jr., Bowen High School, Chicago, who supervised the preparation of drawings for the diagrams and figures; and to many firms and individuals that have courteously furnished material for illustrations.

Willis E. Tower.
Charles H. Smith.
Charles M. Turton.

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[Pg ix]

When a pupil begins the study of Physics he has in his possession many bits of knowledge which are fundamental in the science. He has learned to throw a ball and can tell how a thrown ball moves. He has drawn out nails with a claw hammer. He has seen wood float and iron sink. He has sucked liquids up through straws. In his mother's kitchen, he sees water as ice, liquid, and steam. On a wintry day he reads the temperature on a thermometer. He sees sparks fly from car wheels when the brakes are applied. He has played with a horseshoe magnet, and has found the north by means of a compass. The telephone, the electric light and the motor he sees, and perhaps uses, many times a day. He dresses before a mirror, focuses his camera, watches the images at a moving picture show, and admires the colors of the rainbow. He has cast stones into water to watch the ripples spread, has shouted to hear the echo, and perhaps plays some musical instrument. These, and a thousand other things, are known to the intelligent and normal boy or girl who has reached the age at which the study of Physics is properly begun.

To a great extent even the terms used in the science are familiar to the beginner. He speaks of the horse-power of an engine, reads kilowatt-hours from the meter in the cellar, and may know that illuminating gas costs one dollar per thousand "cubic feet." "Ampere" and "volt" are words he frequently hears and sees.

When he takes up the study of Physics, the attitude of[Pg x] the student toward these familiar things and words must undergo a change. Casual information about them must be changed to sound knowledge, purposely acquired. Hazy notions about the meanings of words must be replaced by exact definitions. Bits of knowledge must be built into a structure in which each fact finds its proper place in relation to the others.

The only agent which can accomplish these changes is the student himself. He must consciously and purposely seek the truth and must reflect upon it until he sees it in its relation to other truth. Upon him, and upon him alone, rests the final responsibility for the success or failure of his study.

But the student is not without assistance. In his teacher he finds a guide to stimulate, to direct, and to aid his efforts, and a critic to point out wherein his efforts have failed and wherein they have succeeded. Weights, measures, and other apparatus are furnished to enable him to answer for himself questions which have arisen in his studies.

In addition to these the student has his text book, his teacher for his hours of private study. A good text book is an inspiring teacher in print. It directs attention to things familiar to the student through long experience, and inspires him to make a closer scrutiny of them. It invites him to observe, to analyze, to compare, to discover likenesses and differences in behavior. It questions him at every turn. Its ever repeated challenge reads, "Weigh and consider." It furnishes him needed information that he cannot otherwise acquire. It satisfies his desire to know, "By whom, where, when, and how was this first discovered?"

The student of Physics must never forget that he is studying not pages of text but the behavior and properties[Pg xi] of iron, water, mica, moving balls, pumps, boiling liquids, compressed air, mirrors, steam engines, magnets, dynamos, violins, flutes, and a host of other things. His studies should, whenever possible, be made first hand upon the things themselves. The text is an aid to study, never a substitute for the thing studied.

It is an excellent plan for each student to select some one thing for special study, the telephone for example. By observation, experiment, and reading, he may acquire a large amount of valuable information about such a subject while pursuing his course in Physics. Every part of the science will be found to bear some relation to it.

The student who takes up the study of Physics in the way suggested will find himself at the end of a year of study in possession of much new and valuable knowledge about the physical world in which he lives. By virtue of this knowledge he will be better able to enjoy the world, to control it, and to use it.

Thomas D. Cope.

Philadelphia.

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[Pg xii]

Chapter I. Introduction And Measurement. Page

Chapter II. Molecular Forces And Motions.

Chapter III. Mechanics or Liquids.

Chapter IV. Mechanics Of Gases.

Chapter V. Force And Motion.

Chapter VI. Work And Energy.

Chapter VII. Heat, Its Production And Transmission.[Pg xiv]

Chapter VIII. Heat And Work.

Chapter IX. Magnetism.

Chapter XI. Static Electricity.

Chapter XI. Electric Currents Produced By Voltaic Cells.

Chapter XII. Magnetic Effects Of Electric Currents, And Electrical Measurements.

Chapter XIII. Chemical And Heat Effects Of Electric Currents.

Chapter XIV. Induced Currents.

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Chapter XV. Sound.

Chapter XVI. Light.

Chapter XVII. Invisible Radiations.

Chapter XVIII. Wireless Telephony And Alternating Currents.

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[Pg 1]

PHYSICS

INTRODUCTION AND MEASUREMENT

1. Physics, an Explanation of Common Things.—Many students take up the study of physics expecting to see wonderful experiments with the "X" rays, wireless telegraphy, dynamos, and other interesting devices. Others are dreading to begin a study that to them seems strange and difficult, because they fear it deals with ideas and principles that are beyond their experience and hard to comprehend.

Each of these classes is surprised to learn that physics is mainly an explanation of common things. It is a study that systematizes our knowledge of the forces and changes about us; such as the pull of the earth, the formation of dew, rain and frost, water pressure and pumps, echoes and music, thermometers and engines, and many other things about us with which people are more or less familiar. Physics is like other school subjects, such as mathematics and language, in having its own peculiar vocabulary and methods of study; these will be acquired as progress is made in the course.

The most useful habit that the student of physics can[Pg 2] form is that of connecting or relating each new idea or fact that is presented to him to some observation or experience that will illustrate the new idea. This relating or connecting of the new ideas to one's own personal experience is not only one of the best known means of cultivating the memory and power of association, but it is of especial help in a subject such as physics, which deals with the systematic study and explanation of the facts of our every-day experience.

2. Knowledge—Common and Scientific.—This leads to the distinction between common knowledge and scientific knowledge. We all possess common knowledge of the things about us, gained from the impressions received by our senses, from reading, and from the remarks of others. Scientific knowledge is attained when the bits of common knowledge are connected and explained by other information gained through study or experience. That is, common knowledge becomes scientific, when it is organized. This leads to the definition: Science is organized knowledge.

Common knowledge of the forces and objects about us becomes scientific only as we are able to make accurate measurements of these. That is, science is concerned not only in how things work, but even more in how much is involved or results from a given activity. For example, a scientific farmer must be able to compute his costs and results in order to determine accurately his net profits. The business man who is conducting his business with efficiency knows accurately his costs of production and distribution.

This book is written in the hope that it will make more scientific the student's common knowledge of the forces and changes in the world about him and will give him many ideas and principles that will help him to acquire the habit of looking from effects to their natural causes and[Pg 3] thus tend to develop what is called the scientific habit of thought.

3. Hypothesis, Theory, and Law.—Three words that are frequently used in science may be mentioned here: hypothesis, theory, and law. An hypothesis is a supposition advanced to explain some effect, change, or condition that has been observed. For example, the Nebular Hypothesis of which many high-school students have heard, is an attempt to explain the origin of the sun, the earth, the planets, and other solar systems.

A theory is an hypothesis which has been tested in a variety of ways and which seems to fit the conditions and results so that it is generally accepted as giving a satisfactory explanation of the matter in question. The Molecular Theory of Matter which states that matter of all kinds is composed of very small particles called molecules (see Art. 6), is a familiar example of a theory.

A theory becomes a law when it may be definitely proved. Many laws are expressed in mathematical language, e.g., the law of gravitation. (See Art. 88.) Many of the laws of physics are illustrated by laboratory experiments, which show in a simple way just what the law means.

Explain what is meant by the following terms and expressions:

1. Common knowledge.

2. Scientific knowledge.

3. Science.

4. Topics in physics.

5. Scientific habit of thought.

6. Value of relating new ideas to former experiences.

7. Hypothesis.

8. Theory.

9. Law.

[Pg 4]

4. Physics Defined.—In the study of any science or field of knowledge, it is helpful to have a basis for grouping or classifying the facts studied. In physics we are to study the objects, forces, and changes about us, to understand them and their relations to one another. Accordingly, physics, dealing with the material world about us, is often defined as the science of matter and energy, matter being anything that occupies space and energy the capacity for doing work. This definition of physics while not strictly accurate is sufficiently comprehensive for our present purpose.

5. The Three States of Matter.—Our bodies are matter since they occupy space. Further, they possess energy since they are able to do work. In beginning the study of physics it will simplify our work if we study one of these topics before the other. We will therefore begin with matter and consider first its three states.

Some bodies are solid; as ice, iron, wax. Others are liquid; as water, mercury, oil. Still others are in the state of gas; as steam, air, and illuminating gas. Further we notice that the same substance may be found in any one of the three states. For example water may be either ice, water or steam; that is, either a solid, a liquid, or a gas.

Most persons have heard of liquid air and possibly some know of ice air, i.e., air cooled until it not only liquefies, but is solidified. On the other hand, iron may be melted and, if heated hot enough, may be turned into iron vapor. In fact most substances by heating or cooling sufficiently may be changed into any one of the three states.

Before defining the three states, let us consider the structure of matter. This may help us to answer the question: How is it possible to change a hard solid, such[Pg 5] as ice, into a liquid, water, and then into an invisible gas like steam? This is explained by the molecular theory of matter.

6. The Molecular Theory of Matter.—It is believed that all bodies are made up of very small particles called molecules, and that these instead of being packed tightly together like square packages in a box, are, strange as it may seem, very loosely packed even in solids and do not permanently touch their neighbors. The size of these molecules is so minute that it has been estimated that if a drop of water could be magnified to the size of the earth, the molecules magnified in the same proportion would be in size between a baseball and a football. The air and all other gases are believed to be made up of molecules in rapid motion, striking and rebounding continually from one another and from any objects in contact with the gas.

7. States of Matter Defined.—These ideas of the structure of matter assist us in understanding the following definitions: A solid is that state of matter in which the molecules strongly cling together and tend to keep the same relative positions. (This of course follows from the tendency of a solid to retain a definite form.) A liquid is that state of matter in which the molecules tend to cling together, yet move about freely. Hence a liquid takes the form of any vessel in which it is placed. A gas is that state of matter in which the molecules move about freely and tend to separate indefinitely. Hence a gas will fill any space in which it is placed.

8. Effect of Heat on Matter.—It is further believed that when a body is heated, that the action really consists in making its molecules move or vibrate faster and faster as the heating progresses. This increase of motion causes the molecules to push apart from one another and this[Pg 6] separation of the molecules causes an expansion of the body whether it be solid, liquid, or gas. Fig. 1 shows the expansion of air in an air thermometer. Fig. 2 shows the expansion of a solid on heating.

9. Physical and Chemical Changes. A change of state such as the freezing or boiling of water is called a physical change, for this change has not affected the identity of the substance. It is water even though it has become solid or gaseous. Heating a platinum wire red hot is also a physical change for the wire when on cooling is found to be the same substance as before. Further if salt or sugar be dissolved in water the act of solution is also a physical change since the identical substance (salt or sugar) is in the solution and may be obtained by evaporating the water.

If some sugar, however, is heated strongly, say in a test-tube, it is found to blacken, some water is driven off and on cooling some black charcoal is found in the tube instead of the sugar. This action which has resulted in a change in the nature of the substance treated is called a chemical change. To illustrate further, if some magnesium wire is heated strongly in a flame, it burns, giving off an intense light and when it cools one finds it changed to a light powdery substance like ashes. Chemical changes,[Pg 7] or those that change the nature of the substance affected, are studied in chemistry. In physics we have to do only with physical changes, that is, with those changes that do not affect the nature of the substance.

1. Physics defined.

2. The three states of matter; solid, liquid, gas.

3. Molecular theory of matter.

4. Physical and chemical changes.

Write out in your own words your understanding of:

1. The structure of matter.

2. Some of the differences between solids, liquids, and gases.

3. How to change solids to liquids and gases and vice versa.

4. The reason for the changes of size of a body on heating.

5. Why cooling a gas tends to change it to a liquid or a solid.

6. The actual size of molecules.

Which of the following changes are chemical and which physical?

Give reasons.

1. Melting of ice.

2. Burning of a candle.

3. Production of steam.

4. Falling of a weight.

5. Drying of clothes.

6. Making an iron casting.

7. Decay of vegetables.

8. Sprouting of seeds.

9. Flying an aeroplane.

10. Growth of a plant.

11. Grinding of grain.

12. Sawing a board.

13. Pulverizing stone.

14. Making toast.

15. Sweetening tea or coffee with sugar.

16. Burning wood or gas.

[Pg 8]

10. The Metric System.—In order to study the three states of matter with sufficient exactness it is necessary to employ a system of measurement. The system universally employed by scientists is called The Metric System. In many respects it is the most convenient for all purposes. Every student should therefore become familiar with it and learn to use it. At the present time, not only do scientists everywhere use it, but many countries have adopted it and use it in common measurements. It was legalized in the United States in 1866. The metric system was originated by the French Academy of Sciences during the latter part of the 18th century. There were so many different systems of weights and measures in use, each country having a system of its own, that commerce was much hindered. It was therefore decided to make a system based upon scientific principles. The length of the earth's quadrant passing from the equator to the pole was determined by surveying and computation. One-ten-millionth of this distance was selected as the unit of length and called a meter. Accurate copies of this meter were made and preserved as standards.

Later surveys have shown that the original determination of the earth's quadrant was not strictly accurate; so that after all the meter is not exactly one-ten-millionth of the earth's quadrant.

11. The Standard Meter.—The standard unit of length in the metric system is the meter. It is the distance, at the temperature of melting ice, between two transverse parallel lines ruled on a bar of platinum (see Fig. 3), which[Pg 9] is kept in the Palace of the Archives in Paris. Accurate copies of this and other metric standards are also kept at the Bureau of Standards at Washington, D. C. Fig. 4 shows the relation between the inch and the centimeter (one-hundredth of a meter).

12. Units and Tables in the Metric System.—The metric unit of area commonly used in physics is the square centimeter.

The standard unit of volume or capacity is the liter. It is a cube one-tenth of a meter on each edge. It is equal to 1.057 quarts. It corresponds, therefore, to the quart in English measure.

The standard unit of mass is the kilogram. It is the mass of 1 liter of pure water at the temperature of its greatest density, 4°C. or 39.2°F.

The three principal units of the metric system, the meter, the liter, and the kilogram, are related to one another in a simple manner, since the liter is a cube one-tenth of a meter in each dimension and the kilogram is the mass of a liter of water. (See Fig. 5.)

The metric system is a decimal system that is, one unit is related to another unit in the ratio of ten or of some power of ten. This is indicated by the following tables:

[Pg 10]

The measures commonly used are the centimeter, meter and kilometer.

The masses commonly used are the milligram, gram and kilogram.

Notice in these tables the similarity to 10 mills equal 1 cent, 10 cents equal 1 dime, 10 dimes equal 1 dollar, in the table of United States money.

Other tables in the metric system are built upon the same plan. Learn the prefixes in order thus: milli, centi, deci, deka, hecto, kilo, myria. The first three prefixes are Latin numerals and represent divisions of the unit. The last four are Greek numerals and represent multiples. In these tables, milli means 1/1000, centi means 1/100, deci means 1/10, deka means 10, hecto, 100, kilo, 1000, myria, 10,000. Two other prefixes are sometimes used, micro which means 1/1,000,000; as microfarad or microvolt, and meg which means 1,000,000, as megohm meaning 1,000,000 ohms.

13. Advantages of the Metric System.—First, it is a decimal system; second, the same form and prefixes are used in every table; third, the standards of length (meter), volume (liter), and mass (kilogram) bear a simple relation[Pg 11] to one another. This simple relation between the three standard units may be given thus: first, the liter is a cubic decimeter, and second, the kilogram is the mass of a liter of water. (See Fig. 5) Since the liter is a cubic decimeter, the length of one side is 10 cm. The liter therefore holds 1000 ccm. (10 × 10 × 10). Therefore, 1 liter = 1 cu. dm. = 1000 ccm. and since 1 liter of water has a mass of 1 kg. or 1000 g., then 1000 ccm. of water has a mass of 1000 g., or 1 ccm. of water has a mass of 1 g.

The following table of equivalents gives the relation between the most common English and metric units. Those marked (*) should be memorized.

The c. g. s. system. Scientists have devised a plan for expressing any measurement in terms of what are called the three fundamental units of length,[Pg 12] mass, and time. The units used are the centimeter, the gram and the second. Whenever a measurement has been reduced to its equivalent in terms of these units, it is said to be expressed in C.G.S. units.

1. The metric system; how originated.

2. Units; meter, liter, kilogram.

3. Metric tables.

4. Advantages of the metric system.

5. Equivalents.

6. The C.G.S. system.

1. Which is cheaper, milk at 8 cents a quart or 8 cents a liter? Why?

2. Which is more expensive, cloth at $1.00 a yard or at $1.00 a meter? Why?

3. Which is a better bargain, sugar at 5 cents a pound or 11 cents a kilogram? Why?

4. Express in centimeters the height of a boy 5 ft. 6 in. tall.

5. What is the length of this page in centimeters? In inches?

6. What is the mass of a liter of water? Of 500 ccm.? Of 1 ccm.?

7. From Chicago to New York is 940 miles. Express in kilometers.

8. A 10-gallon can of milk contains how many liters?

9. What will 100 meters of cloth cost at 10 cents a yard?

10. What will 4 kg. of beef cost at 15 cents a pound?

11. What will 5-1/2 lbs. of mutton cost at 40 cents a kilogram?

12. How can you change the state of a body? Give three methods.

13. Correct the statement 1 ccm. = 1 g.

14. How many liters in 32 quarts?

[Pg 13]

MOLECULAR FORCES AND MOTIONS

14. Size of Molecules.—The difference between solids, liquids, and gases has been explained as due to the different behavior of molecules in the three states of matter. That is, in solids they cling together, in liquids they move freely, and in gases they separate. At this time we are to consider the evidences of molecular motion in gases. It must be kept in mind that molecules are exceedingly small. It has been said that if a bottle containing about 1 ccm. of ordinary air has pierced in it a minute opening so that 100,000,000 molecules (a number nearly equal to the population of the United States) pass out every second, it would take, not minutes or hours, but nearly 9000 years for all of the molecules to escape. The number of molecules in 1 ccm. of air at 0°C. and 76 cm. pressure has been calculated by Professor Rutherford to be 2.7 × 10¹⁹. It is evident that such minute particles cannot be seen or handled as individuals. We must judge of their size and action by the results obtained from experiments.

15. Diffusion of Gases.—One line of evidence which indicates that a gas consists of moving particles is the rapidity with which a gas having a strong odor penetrates to all parts of a room. For example, if illuminating gas is escaping it soon diffuses and is noticed throughout the room. In fact, the common experience of the diffusion of gases having a strong odor is such that we promptly[Pg 14] recognize that it is due to motion of some kind. The gas having the odor consists of little particles that are continually hitting their neighbors and are being struck and buffeted in turn until the individual molecules are widely scattered. When cabbage is boiled in the kitchen soon all in the house know it. Other illustrations of the diffusion of gases will occur to anyone from personal experience, such for instance as the pleasing odor from a field of clover in bloom.

The following experiment illustrates the rapid diffusion of gases.

On account of their small size, molecules of air readily pass through porous solids, cloth, unglazed earthenware, etc. The following experiment shows this fact strikingly. (See Fig. 6b.)

[Pg 15]

Careful experiments have shown that the speed of ordinary air molecules is 445 meters or 1460 ft. per second; while hydrogen molecules move at the rate of 1700 meters or 5575 ft. or more than a mile per second.

16. Expansion of Gases.—Gases also possess the property of indefinite expansion, that is, if a small quantity of gas is placed in a vacuum, the gas will expand immediately to fill the entire space uniformly. This is shown by an experiment with the air pump. On raising the piston the air follows instantly to fill up the space under it. As[Pg 16] the air is removed from the receiver of an air pump the air remaining is uniformly distributed within.

17. How Gases Exert Pressure.—It is further found that air under ordinary conditions exerts a pressure of about 15 lbs. to the square inch. In an automobile tire the pressure may be 90 lbs. and in a steam boiler it may be 200 lbs. or more to the square inch.

How is the pressure produced? The molecules are not packed together solidly in a gas, for when steam changes to water it shrinks to about 1/1600 of its former volume. Air diminishes to about 1/800 of its volume on changing to liquid air. The pressure of a gas is not due then to the gas filling all of the space in which it acts, but is due rather to the motion of the molecules. The blow of a single molecule is imperceptible, but when multitudes of molecules strike against a surface their combined effect is considerable. In fact, this action is known to produce the pressure that a gas exerts against the walls of a containing vessel. Naturally if we compress twice as much gas into a given space there will be twice as many molecules striking in a given time, which will give twice as much pressure.

If gas is heated, it is found that the heat will cause a swifter motion of the molecules. This will also make the molecules strike harder and hence cause the gas to expand or exert more pressure.

17a. Brownian Movements.—Direct photographic evidence of the motion of molecules in gases has been obtained by studying the behavior of minute drops of oil suspended in stagnant air. Such drops instead of being at rest are constantly dancing about as if they were continually receiving blows from many directions. These motions have been called Brownian Movements (see Fig. 7).

[Pg 17]

It has been proved that these movements are due to the blows that these small drops receive from the swiftly moving molecules of the gas about them. If the drops are made smaller or the gas more dense, the movements increase in intensity. These effects are especially marked at a pressure of 0.01 of an atmosphere.

It is assumed that air and all gases are made up of molecules in rapid motion; that this motion is dependent upon temperature and pressure. Evidence of this is shown by (a) diffusion, (b) expansion, (c) pressure. Brownian Movements.

1. What is the molecular (kinetic) theory of gases?

2. What three kinds of evidence help to confirm the theory?

3. What have you seen that seems to show that a gas consists of molecules in motion?

4. How many meters long is a 10-ft. pole?

5. A 50-kg. boy weighs how many pounds?

6. What are three advantages of the metric system?

7. What will 12 qts. of milk cost at 8 cents a liter?

8. A cube 1 meter each way will contain how many cubic centimeters? How many liters? What will a cubic meter of water weigh?

[Pg 18]

18. Diffusion of Liquids.—From the evidence given in Arts. 14-17, (a) of diffusion of odors, (b) of the continued expansion of air in the air pump, and (c) of the pressure exerted by a gas in all directions, one may realize without difficulty that a gas consists of small particles in rapid motion. Let us now consider some of the evidence of molecular motion in liquids. If a little vinegar is placed in a pail of water, all of the water will soon taste sour. A lump of sugar in a cup of tea will sweeten the entire contents. This action is somewhat similar to the diffusion of gases but it takes place much more slowly. It is therefore believed that the motion of liquid molecules is much slower than that of gas molecules.

Again, if a dish of water is left standing in the open air in fine weather, within a few days the dish will become dry though no one has taken anything from it. We say the water has evaporated. What was liquid is now vapor. If we were to observe carefully any dish of water we would find that it continually loses weight on dry days. That is, there is a constant movement of the molecules of water into the air. This movement of the molecules is explained as follows. There appear to be in the dish of water some molecules that by moving back and forth acquire a greater velocity than their neighbors; when these reach the surface of the liquid, some vibration or movement sends them flying into the air above. They are now vapor or gas molecules, flying, striking, and rebounding like the air molecules. Sometimes on rebounding, the water molecules get back into the water again. This is especially apt to happen when the air is damp, i.e., when it contains many water molecules. Sometimes the air over a dish becomes saturated, as in the upper part of a corked bottle[Pg 19] containing water. Although molecules are continually leaving the surface of the water they cannot escape from the bottle, so in time as many molecules must return to the water from the space above as leave the water in the same time. When this condition exists, the air above the water is said to be saturated. On very damp days the air is often saturated. The explanation above shows why wet clothes dry so slowly on such a day (See Arts. 166-7 on Saturation.)

19. Cooling Effect of Evaporation. We have seen that warming a gas increases its volume. This expansion is due to the increased motion of the warmed molecules. Now the molecules that escape from a liquid when it evaporates are naturally the fastest moving ones, i.e., the hottest ones. The molecules remaining are the slower moving ones or colder molecules. The liquid therefore becomes colder as it evaporates, unless it is heated. This explains why water evaporating on the surface of our bodies cools us. In evaporating, the water is continually losing its warm, fast moving molecules. The cooling effect of evaporation is, therefore an evidence of molecular motion in liquids.

20. Osmosis.—If two liquids are separated by a membrane or porous partition, they tend to pass through and mix. This action is called osmose, or osmosis.

Gases and liquids are alike in that each will flow. Each is therefore called a fluid. Sometimes there is much resistance to the flow of a liquid as in molasses. This resistance is called viscosity. Alcohol and gasoline have little viscosity. They are limpid or mobile. Air also has some viscosity. For instance, a stream of air always drags some of the surrounding air along with it.

1. Liquids behave as if they were composed of small particles in motion.

2. This is shown by (1) Diffusion, (2) Solution, (3) Evaporation, (4) Expansion, (5) Osmosis.

1. Give an example or illustration of each of the five evidences of molecular motion in liquids.

2. When is air saturated? What is the explanation?

3. Why does warming a liquid increase its rate of evaporation?

4. Air molecules are in rapid motion in all directions. Do they enter a liquid with a surface exposed to the air? Give reason.

5. What are some of the inconveniences of living in a saturated atmosphere?

6. Fish require oxygen. How is it obtained?

[Pg 21]

21. Cohesion and Adhesion.—In liquids "the molecules move about freely yet tend to cling together." This tendency of molecules to cling together which is not noticeable in gases is characteristic of liquids and especially of solids. It is the cause of the viscosity mentioned in the previous section and is readily detected in a variety of ways. For instance, not only do liquid molecules cling together to form drops and streams, but they cling to the molecules of solids as well, as is shown by the wet surface of an object that has been dipped in water. The attraction of like molecules for one another is called cohesion, while the attraction of unlike molecules is called adhesion, although the force is the same whether the molecules are alike or unlike. It is the former that causes drops of water to form and that holds iron, copper, and other solids so rigidly together. The adhesion of glue to other objects is well known. Paint also "sticks" well. Sometimes the "joint" where two boards are glued together is stronger than the board itself. The force of attraction between molecules has been studied carefully. The attraction acts only through very short distances. The attraction even in liquids is considerable and may be measured. The cohesion of water may be shown by an experiment where the force required to pull a glass plate from the surface of water is measured.

[Pg 22]

Just as the plate comes from the water its under side is found to be wet. That is, the water was pulled apart, and the plate was not pulled from the water. The cohesion of the water to itself is not so strong as its adhesion to the glass.

The cohesion of liquids is further shown by the form a drop of liquid tends to take when left to itself. This is readily seen in small drops of liquids. The spherical shape of drops of water or mercury is an example. A mixture of alcohol and water in proper proportions will just support olive oil within it. By carefully dropping olive oil from a pipette into such a mixture, a drop of the oil, an inch or more in diameter suspended in the liquid, may be formed. It is best to use a bottle with plane or flat sides, for if a round bottle is used, the sphere of oil will appear flattened.

22. Surface Tension.—The cohesion of liquids is also indicated by the tendency of films to assume the smallest possible surface. Soap bubble films show this readily. Fig. 10 a represents a circular wire form holding a film in which floats a loop of thread. The tension of the[Pg 23] film is shown in Fig. 10 b by the circular form of the loop after the film within it has been pierced by a hot wire, Fig. 11 shows a rectangular wire form with a "rider." The tension in the film draws the rider forward.

A soap bubble takes its spherical shape because this form holds the confined air within the smallest possible surface. A drop of liquid is spherical for the same reason. Many illustrations of the tension in films may be given. Users of water colors notice that a dry camel's-hair brush is bushy. (Fig. 12 A). When in water it is still bushy. (Fig. 12 B.) But when it is taken from the water and the excess is shaken from it, it is pointed as in Fig. 12 C. It is held to the pointed shape by the tension of the liquid film about the brush.

The surface of water acts as if covered by a film which coheres more strongly than the water beneath it. This is shown by the fact that a steel needle or a thin strip of metal may be floated upon the surface of water. It is supported by the surface film. (See Fig. 13.) If the film breaks the needle sinks. This film also supports the little water bugs seen running over the surface of a quiet pond in[Pg 24] summer. The surface film is stronger in some liquids than in others. This may be shown by taking water, colored so that it can be seen, placing a thin layer of it on a white surface and dropping alcohol upon it. Wherever the alcohol drops, the water is seen to pull away from it, leaving a bare space over which the alcohol has been spread. This indicates that the alcohol has the weaker film. The film of greasy benzine is stronger than the film of the pure material. If one wishes to remove a grease spot and places pure benzine at the center of the spot, the stronger film of the greasy liquid will pull away from the pure benzine, and spread out, making a larger spot than before, while if pure benzine is placed around the grease spot, the greasy liquid at the center pulls away from the pure benzine, drawing more and more to the center, where it may be wiped up and the grease entirely removed.

23. Explanation of the Surface Film.—Beneath the surface of a liquid each molecule is attracted by all the other molecules around it. It is attracted equally in all directions. Consequently the interior molecules move very easily over each other in any direction. A molecule at the surface, as at A, Fig. 14, is not attracted upward by other liquid molecules. Its freedom of motion is thereby hindered with the result that a molecule at the surface[Pg 25] behaves differently from one beneath the surface. The surface molecules act as if they form an elastic skin or membrane upon the liquid surface.

24. Capillarity.—A striking action of the surface film of a liquid is seen in the rise of liquids in tubes of small bore when the liquid wets them. If the liquid does not wet the tube, as when mercury is placed in glass, the liquid is depressed. It is found in general that: Liquids rise in capillary tubes when they wet them and are depressed in tubes which they do not wet; the smaller the diameter of the tube the greater the change of level. (See Fig. 15.) This action is explained as follows: The molecules of a liquid have an attraction for each other and also for the sides of a tube. The former is called "cohesion for itself," the latter is called "adhesion for the sides of the containing vessel." If the cohesion for itself is greater than the adhesion for the side of the containing vessel, the liquid is pulled away from the side and is depressed. If the adhesion is greater, the liquid is elevated. This action is called "capillary action" from the Latin word (capillus) signifying hair, since it shows best in fine hairlike tubes.

There are many common illustrations of capillary action: oil rising in a wick; water rising in a towel or through clothes; ink in a blotter, etc. The minute spaces between the fibers composing these objects act as fine tubes. If cloth is treated with a preparation which prevents water from adhering to its fibers, the material will not be wet when water is poured upon it, because[Pg 26] the water will not run in between the fibers; a surface film spreads over the cloth so that no water enters it. Cravenette cloth has been treated in this way and hence is waterproof.

25. Capillary Action in Soils.—The distribution of moisture in the soil depends largely upon capillary action. When the soil is compact the minute spaces between the soil particles act as capillary tubes, thus aiding the water to rise to the surface. As the water evaporates from the surface more of it rises by capillary action from the damper soil below. Keeping the soil loose by cultivation, makes the spaces between the particles too large for much capillary action, thus the moisture is largely prevented from rising to the surface.

In the semi-arid regions of the West "dry farming" is successfully practised. This consists in keeping the surface covered with a "dust mulch" produced by frequent cultivation. In this way the moisture is kept below the surface, where it can be utilized during the hot dry summer by the roots of growing plants.

1. Attractive forces between liquid molecules.

2. Cohesion (like molecules); adhesion (unlike molecules).

3. Special effects of this force are classified as (a) capillary action, and (b) surface tension.

[Pg 27]

1. What evidence of capillary action have you seen outside of the laboratory?

2. What is the explanation for capillary action?

3. Where are surface films found?

4. What are three common effects of surface films?

5. Explain why cravenette cloth sheds water.

6. If a circular glass disc 10 cm. in diameter requires 50 grams of force to draw it from the water, what is the cohesion of water per square centimeter?

7. What is the weight in grams of 1 ccm. of water? of a liter of water?

8. Name five examples of adhesion to be found in your home.

9. Under what conditions will a liquid wet a solid and spread over it?

10. When will it form in drops on the surface?

11. Explain the proper procedure for removing a grease spot with benzine.

12. What difference is there between a liquid and a fluid?

13. Why cannot a "soap bubble" be blown from pure water?

14. Which are larger, the molecules of steam or those of water? Why?

15. Why is the ground likely to be damp under a stone or board when it is dry all around?

16. Why does any liquid in falling through the air assume the globule form?

17. Give three examples of capillary attraction found in the home. Three out of doors.

18. Why does cultivation of the soil prevent rapid evaporation of water from the ground?

26. Solutions.—A crystal of potassium permanganate is placed in a liter of water. It soon dissolves and on shaking the flask each portion of the liquid is seen to be colored red. The dissolving of the permanganate is an illustration[Pg 28] of the attraction of the molecules of water for the molecules of the permanganate. We are familiar with this action in the seasoning of food with salt and sweetening with sugar.

Water will dissolve many substances, but in varying degrees, i.e., of some it will dissolve much, of others, little, and some not at all. Further, different liquids have different solvent powers. Alcohol will dissolve resin and shellac, but it will not dissolve gum arabic, which is soluble in water. Benzine dissolves grease. Beeswax is not dissolved by water, alcohol or benzine, but is soluble in turpentine.

It is found that the temperature of the liquid has a marked effect upon the amount of substance that will dissolve. This is an indication that the motions of the molecules are effective in solution. It appears that dissolving a solid is in some respects similar to evaporation, and just as at higher temperatures more of the liquid evaporates, because more of the molecules will escape from the liquid into the air above, so at higher temperatures, more molecules of a solid will detach themselves through greater vibration and will move into the liquid.

Further, just as an evaporating liquid may saturate the space above it so that any escape of molecules is balanced by those returning, so with a dissolving solid, the liquid may become saturated so that the solution of more of the solid is balanced by the return of the molecules from the liquid to the solid condition.

27. Crystals and Crystallization.—This return from the liquid to the solid state, of molecules that are in solution, is especially noticeable when the solution is cooling or evaporating and hence is losing its capacity to hold so much of the solid. On returning to the solid, the molecules attach themselves in a definite manner to the solid[Pg 29] portion, building up regular solid forms. These regular forms are crystals. The action that forms them is called crystallization.

Each substance seems to have its own peculiar form of crystal due to the manner in which the molecules attach themselves to those previously in place. The largest and most symmetrical crystals are those in which the molecules are deposited slowly with no disturbance of the liquid. Beautiful crystals of alum may be obtained by dissolving 25 g. of alum in 50 ccm. of hot water, hanging two or three threads in the solution and letting it stand over night. The thread fibers provide a foundation upon which crystals grow.

When a solution of a solid evaporates, the molecules of the liquid escape as a gas, the molecules of the solid remain accumulating as crystals. This principle has many uses: (a) sea water is purified by evaporating the water and condensing the vapor, which of course forms pure water. (b) water is forced down to salt beds where it dissolves the salt. The brine is then raised and evaporated, leaving the salt in the evaporating pans.

This experiment indicates that attraction exists between gas molecules and other molecules. Many porous substances have this power of absorbing gases. We have all noticed that butter has its flavor affected by substances placed near it.

That liquids absorb gases is shown by slowly heating[Pg 30] cold water in a beaker. Small bubbles of air form on the sides and rise before the boiling point is reached. Ammonia gas is readily absorbed in water, the bubbles disappearing almost as soon as they escape into the water from the end of the delivery tube. Household ammonia is simply a solution of ammonia gas in water. On warming the solution of ammonia the gas begins to pass off; thus, warming a liquid tends to drive off any gas dissolved in it.

Soda water is made by forcing carbon dioxide gas into water under strong pressure. When placed in a vessel open to the air the pressure is lessened and part of the gas escapes. The dissolved gas gives the characteristic taste to the beverage.

1. The solution of solids is increased by heating.

2. The solution of gases is decreased by heating.

3. Pressure increases the quantity of gas that can be dissolved in a liquid.

4. The attraction (cohesion) of molecules of a dissolved solid for each other is shown by crystallization.

1. How do fish obtain oxygen for breathing?

2. Why does warming water enable it to dissolve more of a salt?

3. Why does warming water lessen the amount of a gas that will stay in solution?

4. Will water absorb gases of strong odor? How do you know?

5. Name three solvents. Give a use for each.

6. What liquids usually contain gases in solution? Name some uses for these dissolved gases.

7. What is the weight of a cubic meter of water?

8. Name three substances obtained by crystallization.

9. How is maple sugar obtained?

10. Name five crystalline substances.

[Pg 31]

29. Differences between Solids and Gases.—In studying gases, it is seen that they behave as if they were composed of small particles in rapid motion, continually striking and rebounding, and separating to fill any space into which they are released. This action indicates that there is practically no attractive force between such molecules.

Between the molecules of a solid, however, the forces of attraction are strong, as is shown by the fact that a solid often requires a great force to pull it apart; some, as steel and iron, show this property in a superlative degree, a high-grade steel rod 1 cm. in diameter requiring nearly 9 tons to pull it apart. Tests show that the breaking strengths of such rods are directly proportional to their areas of cross-section. That is, twice the area has twice the breaking strength.

30. Elasticity.—Fully as important as a knowledge of the breaking strengths of solids, is the knowledge of what happens when the forces used are not great enough to break the rods or wires.

This simple experiment illustrates a characteristic of solids: that of changing shape when force is applied and of[Pg 32] returning to the original shape when the force is removed. This property is called elasticity.

Tests of elasticity are made by subjecting wire of different materials but of the same dimensions to the same tension. The one changing least is said to have the greatest elastic force or elasticity. If greater forces are applied to the wire and then removed, one will finally be found that will permanently stretch the wire so that it will not return exactly to the former length. The wire has now passed its elastic limit and has been permanently stretched.

Just as there are great differences between the elastic forces of different substances, so there are great differences in the limits of elasticity. In some substances the limit is reached with slight distortion, while others are perfectly elastic even when greatly stretched. India rubber is an example of a body having perfect elasticity through wide limits. Glass has great elastic force but its limit of elasticity is soon reached. Substances like India rubber may be said to have great "stretchability," but little elastic force. In physics, elasticity refers to the elastic force rather than to ability to endure stretching.

31. Kinds of Elasticity.—Elasticity may be shown in four ways: compression, bending or flexure, extension or stretching, twisting or torsion. The first is illustrated by squeezing a rubber eraser, the second by an automobile spring, the third by the stretching of a rubber band, the fourth by the twisting and untwisting of a string by which a weight is suspended.

There are two kinds of elasticity: (1) elasticity of form or shape; (2) elasticity of volume. Gases and liquids possess elasticity of volume, but not of shape, while solids may have both kinds. Gases and liquids are perfectly elastic because no matter how great pressure may be applied, as soon as the pressure is removed they regain[Pg 33] their former volume. No solid possesses perfect elasticity, because sooner or later the limit of elasticity will be reached.

32. Hooke's Law.[A]—On examining the successive movements of the end of the rod in Art. 30, we find that they are approximately equal. Carefully conducted experiments upon the elasticity of bodies have shown that the changes in shape are directly proportional to the forces applied, provided that the limit of elasticity is not reached. This relation, discovered by Robert Hooke, is sometimes expressed as follows: "Within the limits of perfect elasticity, all changes of size or shape are directly proportional to the forces producing them."

33. Molecular Forces and Molecular Motions.—If a solid is compressed, on releasing the pressure the body regains its former shape if it has not been compressed too far. This indicates that at a given temperature the "molecules of a solid tend to remain at a fixed distance from each other, and resist any attempt to decrease or increase this distance." This raises the question, Why does not the cohesion pull the molecules tightly together so that compression would be impossible? The reason is that heat affects the size of solid bodies. On lowering the temperature, bodies do contract, for as soon as the temperature is lowered the vibration of the molecule is lessened. On raising the temperature the molecules are pushed farther apart.

The size of a body, then, is the result of a balance of opposing forces. The attractive force between the molecules pulling them together is cohesion, while the force which pushes them apart is due to the motions of the molecules. Raising the temperature and thus increasing[Pg 34] the motion causes expansion; lowering the temperature decreases the molecular motion and so causes contraction. If an outside force tries to pull the body apart or to compress it this change of size is resisted by either cohesion or molecular motion.

34. Properties of Matter.—Many differences in the physical properties of solids are due to differences between the cohesive force of different kinds of molecules. In some substances, the attraction is such that they may be rolled out in very thin sheets. Gold is the best example of this, sheets being formed 1/300,000 of an inch thick. This property is called malleability. In other substances the cohesion permits it to be drawn out into fine threads or wire. Glass and quartz are examples of this. This property is called ductility. In some, the cohesion makes the substance excessively hard, so that it is difficult to work or scratch its surface. The diamond is the hardest substance known. Some substances are tough, others brittle. These are tested by the ability to withstand sudden shocks as the blow of a hammer.

1. Molecular forces in solids; (a) adhesion, (b) cohesion.

2. Elasticity, Hooke's Law.

3. Contraction on cooling.

4. Malleability, ductility, hardness, brittleness, etc.

1. Give an illustration of Hooke's Law from your own experience.

2. What devices make use of it?

3. Do solids evaporate. Give reasons.

4. When iron is welded, is cohesion or adhesion acting?

5. When a tin basin is soldered, is cohesion or adhesion acting?

6. Sometimes a spring is made more elastic by tempering and made soft by annealing. Look up the two terms. How is each accomplished?

[Pg 35]

7. Review the definitions: solid, liquid, and gas. Why do these definitions mean more to you now than formerly?

8. If a wire is stretched 0.3 cm. on applying 4 kg. of force, what force will stretch it 0.75 cm? Explain.

9. How long will it take under ordinary conditions for a gas molecule to cross a room? Give reasons for your answer.

10. What is meant by the elastic limit of a body?

11. Without reaching the elastic limit, if a beam is depressed 4 mm. under a load of 60 kg., what will be the depression under a load of 400 kg.? Of 600 kg.?

12. Name three substances that possess elasticity of volume.

13. Give three examples of each; elasticity of (1) compression, (2) stretching, (3) torsion, (4) flexure.

Physics; definition, topics considered, physical and chemical changes.

Science; hypothesis, theory, law. Knowledge; common, scientific.

Matter; three states, molecular theory. Mass, weight, volume.

Metric system; units, tables, equivalents, advantages.

Evidences of molecular motions; gases (3), liquids (5), solids (3).

Evidences of molecular forces; liquids (3), solids (many) special properties such as: elasticity, tenacity, ductility, hardness, etc.

Hooke's law; applications.

[Pg 36]

MECHANICS OF LIQUIDS

35. Pressure of Liquids against Surfaces.—The sight of a great ship, perhaps built of iron and floating on water, causes one to wonder at the force that supports it. This same force is noticed when one pushes a light body, as a cork, under water. It is quite evident in such a case that a force exists sufficient to overcome the weight of the cork so that it tends to rise to the surface. Even the weight of our bodies is so far supported by water that many persons can float.

The following experiment provides a means of testing this force:

36. Law of Liquid Pressure.—As there is twice as much water or shot in the chimney when it is filled to a depth of 10 cm. as there is when it is filled to a depth of 5 cm. the force of the water upward on the bottom must be twice as great at a depth of 10 cm. as at a depth of 5 cm. Since this reasoning will hold good for a comparison of forces at any two depths, we have the law: "The pressure exerted by a liquid is directly proportional to the depth."

The amount of this force may be computed as follows: First, the card stays on the end of the tube until the weight of water from above equals the force of the water from below, and second, the card remains until the water is at the same height inside the tube as it is outside. Now if we find the weight of water at a given depth in the tube, we can determine the force of the water from below. If for instance the chimney has an area of cross-section of 12 sq. cm. and is filled with water to a depth of 10 cm., the volume of the water contained will be 120 ccm. This volume of water will weigh 120 g. This represents then, not only the weight of the water in the tube, but also the force of the water against the bottom. In a similar way one may measure the force of water against any horizontal surface.

37. Force and Pressure.—We should now distinguish between force and pressure. Pressure refers to the force acting against unit area, while force refers to the action against the whole surface. Thus for example, the atmospheric pressure is often given as 15 pounds to the square inch or as one kilogram to the square centimeter. On[Pg 38] the other hand, the air may exert a force of more than 300 pounds upon each side of the hand of a man; or a large ship may be supported by the force of thousands of tons exerted by water against the bottom of the ship.

In the illustration, given in Art. 36, the upward force of the water against the end of the tube at a depth of 10 cm. is computed as 120 grams. The pressure at the same depth will be 10 grams per sq. cm. What will be the pressure at a depth of 20 cm.? at a depth of 50 cm.? of 100 cm.? Compare these answers with the law of liquid pressure in Art. 36.

38. Density.—If other liquids, as alcohol, mercury, etc., were in the jar, the chimney would need filling to the same level outside, with the same liquid, before the card would fall off. This brings in a factor that was not considered before, that of the mass[B] of a cubic centimeter of the liquid. This is called the density of the liquid. Alcohol has a density of 0.8 g. per cubic centimeter, mercury of 13.6 g. per cubic centimeter, while water has a density of 1 g. per cubic centimeter.

39. Liquid Force against Any Surface.—To find the force exerted by a liquid against a surface we must take into consideration the area of the surface, and the height and the density of the liquid above the surface. The following law, and the formula representing it, which concisely expresses the principle by which the force exerted by a liquid against any surface may be computed, should be memorized:

Or, expressed by a formula, F = Ahd. In this formula,[Pg 39] "F" stands for the force which a liquid exerts against any surface, "A" the area of the surface, "h," for the average depth (or height) of the liquid pressing on the surface, and "d", for the weight of unit volume of the liquid. This is the first illustration in this text, of the use of a formula to represent a law. Observe how accurately and concisely the law is expressed by the formula. When the formula is employed, however, we should keep in mind the law expressed by it.

We must remember that a liquid presses not only downward and upward but sideways as well, as we see when water spurts out of a hole in the side of a vessel. Experiments have shown that at a point the pressure in a fluid is the same in all directions, hence the rule given above may be applied to the pressure of a liquid against the side of a tank, or boat, or other object, provided we are accurate in determining the average depth of the liquid; The following example illustrates the use of the law.

1. Liquids exert pressure; the greater the depth the greater the pressure.

[Pg 40]

2. Difference between force and pressure.

3. Rules for finding upward and horizontal force exerted by a liquid. F = Ahd.

4. Weight, mass, density.

1. What is the density of water?

2. What force is pressing upward against the bottom of a flat boat, if it is 60 ft. long, 15 ft. wide and sinks to a depth of 2 ft. in the water? What is the weight of the boat?

3. If a loaded ship sinks in the water to an average depth of 20 ft., the area of the bottom being 6000 sq. ft., what is the upward force of the water? What is the weight of the ship?

4. If this ship sinks only 10 ft. when empty, what is the weight of the ship alone? What was the weight of the cargo in Problem 3?

5. What is the liquid force against one side of an aquarium 10 ft. long, 4 ft. deep and full of water?

6. What is the liquid force on one side of a liter cube full of water? Full of alcohol? Full of mercury? What force is pressing on the bottom in each case?

7. What depth of water will produce a pressure of 1 g. per square centimeter? 10 g. per square centimeter? 1000 g. per square centimeter?

8. What depth of water will produce a pressure of 1 lb. per square inch? 10 lbs. per square inch? 100 lbs. per square inch?

9. What will be the force against a vertical dam-breast 30 meters long, the depth of the water being 10 meters?

10. A trap door with an area of 100 sq. dcm. is set in the bottom of a tank containing water 5 meters deep. What force does the water exert against the trap door?

11. What is the force on the bottom of a conical tank, filled with water, the bottom of which is 3 meters in diameter, the depth 1.5 meters?

12. If alcohol, density 0.8 were used in problem 11, what would be the force? What would be the depth of alcohol to have the same force on the bottom as in problem 11?

13. What is the pressure in pounds per square inch at a depth of 1 mile in sea water, density 1.026 grams per cc.?

[Pg 41]

14. Find the force on the sides and bottom of a rectangular cistern filled with water, 20 ft. long, 10 ft. wide, and 10 ft. deep?

15. Find the force on the bottom of a water tank 14 ft. in diameter when the water is 15 ft. deep, when full of water.

16. Find the force on one side of a cistern 8 ft. deep and 10 ft. square, when full of water.

17. Find the force on a vertical dam 300 ft. long and 10 ft. high, when full of water.

18. Find the pressure at the bottom of the dam in question 17.

19. Why are dams made thicker at the bottom than at the top?

20. A ship draws 26 ft. of water, i.e., its keel is 26 ft. under water. What is the liquid force against a square foot surface of the keel? Find the pressure on the bottom.

40. Pascal's Principle.—Liquids exert pressure not only due to their own weight, but when confined, may be made to transmit pressure to considerable distances. This is a matter of common knowledge wherever a system of waterworks with connections to houses is found, as in cities. The transmission of liquid pressure has a number of important applications. The principle underlying each of these was first discovered by Pascal, a French scientist of the seventeenth century. Pascal's Principle, as it is called, may be illustrated as follows:

[Pg 42]

The usual form in which this law is expressed is as follows: Pressure applied to any part of a confined liquid is transmitted unchanged, in all directions, and adds the same force to all equal surfaces in contact with the liquid.

The importance of this principle, as Pascal himself pointed out, lies in the fact that by its aid we are able to exert a great force upon a large area by applying a small force upon a small area of a confined liquid, both areas being in contact with the same liquid. Thus in Fig. 19 if the area of the surface CD is 2000 times the area of the surface AB, then 1 lb. applied to the liquid on AB will exert or sustain a force of 2000 lbs. on CD.

41. Hydraulic Press.—An important application of Pascal's principle is the hydraulic press. See Fig. 20. It is used for many purposes where great force is required, as in pressing paper or cloth, extracting oil from seeds, lifting heavy objects, etc. Many high school pupils have been seated in a hydraulic chair used by a dentist or barber. This chair is a modified hydraulic press.

[Pg 43]

The hydraulic press contains two movable pistons, P and p (see Fig. 20). The larger of these, P, has a cross-sectional area that may be 100 or 1000 times that of the smaller. The smaller one is moved up and down by a lever; on each upstroke, liquid is drawn in from a reservoir, while each down-stroke forces some of the liquid into the space about the large piston. Valves at V and V´ prevent the return of the liquid. If the area of P is 1,000 times that of p, then the force exerted by P is 1000 times the force employed in moving p. On the other hand, since the liquid moved by the small piston is distributed over the area of the large one, the latter will move only 1/1000 as far as does the small piston. The relation between the motions of the two pistons and the forces exerted by them may be stated concisely as follows: The motions of the two pistons of the hydraulic press are inversely proportional to the forces exerted by them. The cross-sectional areas of the two pistons are, on the other hand, directly proportional to the forces exerted by them.

An application of Pascal's principle often employed in[Pg 44] cities is the hydraulic elevator. In this device a long plunger or piston extends downward from the elevator car into a cylinder sunk into the earth, sometimes to a depth of 300 ft. Water forced into this cylinder pushes the piston upward and when the water is released from the cylinder the piston descends.

42. Artesian Wells.—Sometimes a porous stratum containing water in the earth's crust is inclined. Then if there are impervious strata (see Fig. 22), both above and below the water-bearing one, and the latter comes to the surface so that rain may fill it, a well sunk to the water-bearing stratum at a point where it is below the surface will usually give an artesian well, that is, one in which the water rises to or above the surface. Many are found in the United States.

43. Standpipes and Air Cushions.—Many who have[Pg 45] lived in cities where water is pumped into houses under pressure know that the water pressure is changed when several faucets are opened at the same time. Again, if several persons are using a hose for sprinkling, the pressure may be lessened so as to be insufficient to force the water above the first floor. In order to allow for these changes some flexibility or spring must be introduced somewhere[Pg 46] into the water-pipe system. Water is nearly incompressible and if no means were employed to take care of the pressure changes, the sudden stopping and starting of the flow would cause serious jars and start leaks in the pipes. Two common devices for controlling sudden changes in the water pressure are the standpipe and the air cushion.

1. Pascal's law.

2. Hydraulic press.

3. Artesian wells.

4. Standpipes and air cushions.

[Pg 47]

1. Where have you seen an air cushion? Describe it and its use.

2. Where have you seen an hydraulic press? Why and how used?

3. Where have you seen hydraulic elevators? What moves them?

4. Where do you know of liquids under pressure? Three examples.

5. What is the pressure in water at a depth of 1500 cm. Express in grams per square centimeter and in kilograms per square centimeter.

6. What head[C] of water is required to give a pressure of 200 g. per square centimeter? 2 kg. per square centimeter?

7. What pressure will be produced by a "head" of water of 20 meters?

8. If 1728 cu. in. of water are placed in a vertical tube 1 sq. in. in cross section to what height would the water rise? It would give how many feet of head?

9. What would the water in problem 8 weigh? What pressure would it produce at the bottom, in pounds per square inch? From this, compute how many feet of "head" of water will produce a pressure of 1 lb. per square inch.

10. Using the result in problem 9, what "head" of water will produce a pressure of 10 lbs. per square inch? 100 lbs. per square inch?

11. From the result in 9, 100 ft. of "head" of water will produce what pressure? 1000 ft. of "head?"

12. If the diameter of the pump piston in a hydraulic press is 2 cm. and that of the press piston 50 cm. what will be the force against the latter if the former is pushed down with a force of 40 kg.?

44. A Body Supported by a Liquid.—Among the applications of the force exerted by a liquid upon a surface, Archimedes' Principle is one of the most important.

Most persons have noted that a body placed in water is partly or wholly supported by the force of the water upon it. A stone held by a cord and lowered into water is felt[Pg 48] to have a part of its weight supported, while a piece of cork or wood is wholly supported and floats.

The human body is almost entirely supported in water, in fact, many people can easily float in water. It was the consideration of this fact that led the Greek philosopher Archimedes to discover and state the principle that describes the supporting of a body in a liquid.

45. Archimedes' Principle.—"A body immersed in a liquid is pushed up by a force equal to the weight of the liquid that it displaces." The proof for this law is simply demonstrated. Suppose a cube, abcd, is immersed in water (Fig. 25). The upward force on cd is equal to the weight of a column of water equal to cdef. (See Art. 39.) The downward force upon the top of the cube is equal to the weight of the column of water abef. Then the net upward force upon the cube, that is, the upward force upon the bottom less the downward force upon the top, or the buoyant force exerted by the liquid is exactly equal to the weight of the displaced water abcd.

46. Law of Floating Bodies.—This same reasoning may be applied to any liquid and to any body immersed to any depth below the surface of the liquid. If the body weighs more than the displaced liquid it will sink. If it weighs less than the displaced liquid it will float or rise in the water. A block of wood rises out of the water in which it floats until its own weight just equals the weight of the water it displaces. From this we have the law of floating bodies.

[Pg 49]

A floating body displaces its own weight of the liquid in which it floats.

To test the law of floating bodies, take a rod of light wood 1 cm. square and 30 cm. long (Fig. 26). Bore out one end and fill the opening with lead and seal with paraffin so that the rod will float vertically when placed in water. Mark upon one side of the rod a centimeter scale, and dip the rod in hot paraffin to make it waterproof. Now find the weight of the stick in grams and note the depth to which it sinks in water in centimeters. Compute the weight of the displaced water. It will equal the weight of the rod.

47. Applications of Archimedes' Principle. There are numerous applications of Archimedes' Principle and the law of floating bodies.

Archimedes' Principle, however, provides a method of finding the volume of a body accurately by weighing it first in air and then in water (Fig. 27), the apparent loss in weight being equal to the weight of the displaced water. One needs only to find the volume of water having the same weight as the loss of weight to find the volume of the body.

If the metric system is used, 1 ccm. of water weighs 1 g., and the volume is numerically the same as the loss of weight.

1. Archimedes' Principle.

2. Law of floating bodies.

3. The applications of Archimedes' Principle are to determine (a) the weight of a floating body; (b) the volume of an immersed solid, and (c) the density of a body.

1. Look up the story of Archimedes and the crown. Write a brief account of it.

2. Why is it easier for a fat man to float in water than for a lean one?

[Pg 51]

3. A fish weighing 1 lb. is placed in a pail full of water. Will the pail and contents weigh more than before adding the fish? Why?

4. Why can a large stone be lifted more easily while under water than when on the land?

5. Why does the air bubble in a spirit level move as one end of the instrument is raised or lowered?

6. Why does a dead fish always float?

7. A ship is built for use in fresh water. What will be the effect on its water line when passing into the ocean?

8. Why can small bugs walk on water while large animals cannot?

9. If an object weighing 62.4 lbs. just floats in water, what weight of water does it displace? What volume of water is displaced? What is the volume of the body?

10. What is the volume of a man who just floats in water if he weighs 124.8 lbs.? If he weighs 187.2 lbs.?

11. An object weighing 500 g. just floats in water. What is its volume? How much water does a floating block of wood displace if it weighs 125 lbs.? 125 g.? 2 kg.? 2000 kg.?

12. A flat boat 10 × 40 ft. in size will sink how much in the water when 10 horses each weighing 1250 lbs. are placed on board?

13. A ship 900 ft. long and 80 ft. average width sinks to an average depth of 25 ft. when empty and 40 ft. when loaded. What is the weight of the ship and of its load?

14. Will a 1000 cc. block sink or float in water if it weighs 800 g.? If it weighs 1200 g.? Explain.

15. If a 1000 cc. block of metal weighing 1200 g. is placed in the water in mid ocean what will become of it?

16. Prove Archimedes' Principle by use of the principles of liquid pressure.

17. An irregular stone, density 2.5 g. per ccm. displaces 2 cu. ft. of water. What is its weight? Its apparent weight in water?

18. Will the depth to which a vessel sinks in water change as she sails from Lake Ontario into the Atlantic Ocean? Why?

19. If the density of sea water is 1.0269 g. per cubic centimeter and that of ice 0.918 g. per ccm., what portion of an iceberg is above water?

20. In drawing water from a well by means of a bucket, why is less force used when it is under water than when entirely above?

[Pg 52]

21. A stone which weighs 300 lbs. can be lifted under water with a force of 150 lbs. What is the volume of the stone?

22. The average density of the human body is 1.07 grams per c.c. How much water will a man who weighs 150 lbs. displace when diving? How much when floating?

48. Density.—The density of a substance is often used as a test of its purity. Archimedes in testing King Hiero's crown to find out if it were made of pure gold determined first its density. It is by such tests that the purity of milk, of alcohol, of gold, and a great variety of substances is often determined.

Knowledge of methods of finding density is of value to everyone and should be included in the education of every student. The density of a substance is the mass of unit volume of the substance. In the metric system, for example, the density of a substance is the mass in grams per 1 ccm. Taking water, 1 ccm. weighs 1 gr. or its density is therefore 1 g. to the cubic centimeter. A cubic centimeter of aluminium weighs 2.7 g. Its density therefore is 2.7 g. per ccm.

49. Specific Gravity.—Specific gravity is the ratio of the weight of any volume of a substance to the weight of an equal volume of water. Its meaning is not quite the same as that of density, since specific gravity is always a ratio, i.e., an abstract number, as 2.7. Density of a substance is a concrete number, as 2.7 grams per ccm. In the metric system the density of water is one gram per cubic centimeter, therefore we have:

Density (g. per ccm.) = (numerically) specific gravity.

In the English system, the density of water is 62.4 pounds per cubic foot, therefore in this system we have:

Density (lbs. per cu. ft.) = (numerically) 62.4 × sp. gr.

[Pg 53]

50. Methods for Finding Density and Specific Gravity

(a) Regular Solids.—Solids of regular shapes such as cubes, spheres, etc., whose volumes may be readily found by measurement, may be weighed. The mass divided by the volume gives the density, or D = Mμ/v.

(b) Irregular Solids.—with these the volume cannot be found by measurement but may be obtained by Archimedes' Principle. Weigh the solid first in the air and then in water. The apparent loss of weight equals the weight of the equal volume of water displaced. From this the volume may be found. And then the density equals mass/volume; the specific gravity =

(c) Solids Lighter than Water.—This will require a sinker to hold the body under water. Weigh the solid in air (w). Weigh the sinker in water (s). Attach the sinker to the solid and weigh both in water (w´). The specific gravity equals

The apparent loss of weight of the solid is equal to the sum of its weight in air plus the weight of the sinker in water, less the combined weight of both in water.

(d) The Density of a Liquid by a Hydrometer.—One may also easily find the density of any liquid by Archimedes' Principle. If one takes the rod described in Art. 46, and places it in water, the number of cubic centimeters of water it displaces indicates its weight in grams. On placing the rod in another liquid in which it floats, it will of course displace its own weight and the height to which the liquid rises on the scale gives the volume. By dividing[Pg 54] the weight of the rod as shown by its position in water by the volume of the liquid displaced we obtain the density of the liquid. Commercial hydrometers for testing the density of milk, alcohol and other liquids are made of glass of the form shown in Fig. 28. The long narrow stem permits small differences in volume to be noticed, hence they are more accurate than the rod described in the preceding paragraph. For convenience this rod contains a paper scale, so that when the height of the liquid on the stem is noted, the density is read at once.

Density of Liquids by Loss of Weight. Weigh a piece of glass in air (W_a), in water (W_w), and in the liquid to be tested (W_l).

Then (W_a - W_w)gives the weight of the water displaced.

And (W_a - W_l) gives the weight of the liquid displaced.

Hence, (W_a - W_l)/(W_a - W_w) equals the specific gravity of the liquid.

1. Definitions of density and specific gravity.

2. Methods of finding density: (a) regular solids; (b) irregular solids; (c) solids lighter than water; (d) liquids by hydrometer; (e) liquids by loss of weight.

Note.—Consider that 1 cu. ft. of water weighs 62.4 lbs. Consider that 1 ccm. of water weighs 1 g.

1. What is meant by the statement that a block of wood has a specific gravity of 0.6?

2. Considering that the density of the human body is the same as that of water, what is the volume of a 125-lb. boy? Of a 250-lb. man? Of a 62.4-lb. boy? What is the volume of your body?

3. How is the weight of large ships found? Give an example.

[Pg 55]

4. Mention three cases where determinations of density are important.

5. A body weighs 40 g. in air, 15 g. in water, 5 g. in an acid. Find (a) the density of the body; (b) its volume; (c) density of the acid.

6. If the specific gravity of a horse is 1, what is the volume of a horse weighing 500 kg.? Of one weighing 1248 lbs.?

7. A weighted wooden box sinks to a depth of 20 cm. in water and 24 cm. in alcohol, and to a depth of 18 cm. in brine. What is the density of the alcohol and of the brine?

8. A glass stopper weighs in the air 25 g., in water 15 g., in oil 18 g. Find the density and volume of the stopper. Find the density of the oil.

9. What would a cubic foot of wood weigh if the specific gravity were 0.5.?

10. The specific gravity of aluminum is 2.7. Find the weight of a cubic foot of it.

11. A block of wood weighs 40 g. A piece of lead appears to weigh 70 g. in water. Both together appear to weigh 60 g. in water. Find the density of the wood.

12. A stone weighs 30 g. in air, 22 g. in water, and 20 g. in salt water. Find the density of the salt water.

13. Will iron sink in mercury? Why?

14. A submarine boat weighing 200 tons must have what volume in order to float?

15. Find the weight of 2 cu. ft. of copper from its density.

16. What is the weight in water of a mass whose specific gravity is 3.3 and whose weight is 50 kg.?

17. A block of granite weighs 1656 lbs.; its volume is 10 cu. ft., what is its density?

18. If the specific gravity of hard coal is 1.75 how would you determine how many tons of coal a bin would hold?

19. A hollow copper ball weighs 2 kg. What must be its volume to enable it to just float in water?

20. A mass having a volume of 100 ccm. and a specific gravity of 2.67 is fastened to 200 ccm. of wood, specific gravity 0.55. What will the combination weigh in water?

21. A block weighing 4 oz. in air is tied to a sinker which appears to weigh 14 oz. in water. Both together appear to weigh 6 oz. in water. What is the specific gravity of the block?

[Pg 56]

MECHANICS OF GASES

51. Weight of Air.—It is said that savages are unaware of the presence of air. They feel the wind and hear and see it moving the leaves and branches of the trees, but of air itself they have little conception.

To ordinary observers, it seems to have no weight, and to offer little resistance to bodies passing through it. That it has weight may be readily shown as follows: (See Fig. 29.) If a hollow metal sphere, or a glass flask, provided with tube and stopcock, be weighed when the stopcock is open, and then after the air has been exhausted from it by an air pump, a definite loss of weight is noticeable.

If the volume of the sphere is known and it is well exhausted of air, a fair approximation of the weight of air may be obtained. Under "standard conditions," which means at the freezing temperature and a barometric pressure of 76 cm., a liter of air weighs 1.293 g. while 12 cu. ft. of air weigh approximately 1 lb.

52. Pressure of Air.—Since air has weight it may be supposed to exert pressure like a liquid. That it does so may be shown in a variety of ways.

[Pg 57]

If a plunger fitting tightly in a glass cylinder be drawn upward, while the lower end of the tube is under water, the water will rise in the tube (Fig. 30). The common explanation of this is that the water rises because of "suction." The philosophers of the ancient Greeks explained it by saying that "nature abhors a vacuum," and therefore the water rises. Neither explanation is correct. It was found in 1640 that water would not rise in a pump more than 32 ft. despite the fact that a vacuum was maintained above the water. Galileo was applied to for an explanation. He said, "evidently nature's horror of a vacuum does not extend above 32 ft." Galileo began tests upon "the power of a vacuum" but dying left his pupil Torricelli to continue the experiment. Torricelli reasoned that if water would rise 32 ft., then mercury, which is 13.6 times as dense as water, would rise about 1/13 as much. To test this, he performed the following famous experiment.

53. Torricelli's Experiment (1643).—Take a glass tube about 3 ft. long, sealed at one end, and fill it with mercury. Close the end with the finger and invert, placing the end closed by the finger under mercury in a dish (Fig. 31). Remove the finger and the mercury sinks until the top of the mercury is about 30 in. above the level of the mercury in the dish. Torricelli concluded that the rise of liquids in exhausted tubes is due to the pressure of the atmosphere acting on the surface of the mercury in the dish.

To test this, place the tube with its mercury upon the plate of an air pump and place a tubulated bell jar over[Pg 58] the apparatus so that the tube projects through a tightly fitting stopper. (See Fig. 32.) If the air pressure is the cause of the rise of mercury in the tube, on removing the air from the bell jar the mercury should fall in the tube. This is seen to happen as soon as the pump is started. It is difficult to remove all the air from the receiver so the mercury rarely falls to the same level in the tube as in the dish. A small tube containing mercury is often attached to air pumps to indicate the degree of exhaustion. Such tubes are called manometers.

54. The Amount of Atmospheric Pressure.—Torricelli's experiment enables us to compute readily the pressure of the atmosphere, since it is the atmospheric pressure that balances the column of mercury in the tube. By Pascal's Law, the pressure of the atmosphere on the surface of the mercury in the dish is transmitted as an exactly equal pressure on the mercury[Pg 59] column in the tube at the same level as the mercury outside.

This pressure, due to the air, must balance the weight of the column of mercury in the tube. It therefore equals the weight of the column of mercury of unit cross-section. The average height of the column of mercury at sea-level is 76 cm. Since the weight of 1 cc. of mercury is 13.6 grams, the pressure inside the tube at the level of the surface of the mercury in the dish is equal to 1 × 76 × 13.6 or 1033.6 g. per square centimeter. Therefore the atmospheric pressure on the surface of the mercury in the dish is 1033.6 g. per square centimeter, approximately 1 kg. per square centimeter or 15 lbs. per square inch.

55. Pascal's Experiment.—Pascal tested in another way the action of atmospheric pressure upon the column of mercury by requesting his brother-in-law, Perrier, who lived near a mountain, to try the experiment on its top. Perrier found that on ascending 1000 meters the mercury fell 8 cm. in the tube. Travelers, surveyors, and aviators frequently determine the altitude above sea-level by reading the barometer, an ascent of 11 meters giving a fall of about 1 mm. in the mercury column, or 0.1 in. for every 90 ft. of ascent.

56. The Barometer.—The modern barometer (Fig. 33), consists of a Torricellian tube properly mounted. Reading a barometer consists in accurately reading the height of the mercury column.[Pg 60] This height varies from 75 to 76.5 cm. or 29 to 30 in. in localities not far from the sea-level. The atmospheric pressure varies because of disturbances in the atmosphere. It is found that these disturbances of the atmosphere pass across the country from west to east in a somewhat regular manner, hence a series of readings of the barometer may give reliable information of the movement of these disturbances and so assist in forecasting the weather. The weather Bureau has observations taken at the same moment at various stations over the country. These observations form the basis for the daily forecast of the weather.

1. Weight and Pressure of air in English and metric units. How shown. Evidences.

2. Work of Galileo, Torricelli, and Perrier.

3. Barometer: construction, action, mercurial, aneroid.

1. Do you think Archimedes' Principle applies to the air? Does Pascal's Law? Why?

2. Find the downward pressure of the mercury in a barometer tube if the cross-section is 1 sq. cm. and the height 75 cm. at the level of the mercury surface in contact with the air. (The density of mercury is 13.6 grams per cc.)

[Pg 62]

3. What is the weight of the air in a room if it is 10 × 8 × 4 meters?

4. What weight of air is in a room 10 × 15 × 10 ft.?

5. When smoke rises in a straight line from chimneys, is it an indication of a high or low barometric pressure? Why?

6. Why does a tumbler filled with water and inverted in a dish with its rim under water remain full?

7. If the barometer tube is inclined the mercury remains at the same horizontal level. How can this be explained?

8. When the mercurial barometer stands at 76 cm., how high would a water barometer stand? Explain.

9. Explain why it is possible for one to suck soda water through a tube?

10. Fill a tumbler with water. Place a sheet of paper over the top and invert. The paper clings to the tumbler and prevents the water from escaping. Explain. (See Fig. 35.)

11. Why must a kerosene oil can have two openings in order to allow the oil to flow freely?

12. Explain the action of the modern drinking fountain (Fig. 36).

57. Effect of Pressure on Liquids and Gases.—Both classes of fluids, liquids and gases, have many characteristics in common. Both are composed of molecules that move freely; hence both flow. At any point within a fluid the pressure is the same in all directions. Archimedes' Principle applies, therefore, to both liquids and gases.

We now come to an important difference between liquids and gases. Liquids are practically incompressible. "So much so, that if water is subjected to a pressure of 3000 kg. per sq. cm., its volume is reduced only about one-tenth." Gases show a very different behavior from liquids on being subjected to pressure. They may readily be compressed to a small fraction of their volume as is noticed on inflating a pneumatic tire. A gas has also the ability to spring back to a larger volume as soon as the pressure is released, as when a cork is driven from a pop gun. Not[Pg 63] only is compressed air able to expand, but air under ordinary conditions will expand if it is released in a space where the pressure is less.

Hollow bodies, animals and plants, are not crushed by atmospheric pressure, because the air and gases contained within exert as much force outward as the air exerts inward.

58. Boyle's Law.—The relation between the volume and pressure of a gas was first investigated by Robert Boyle in the seventeenth century. The experiment by which he first discovered the law or the relation between the volume and the pressure of a gas is briefly described as follows:

Thus doubling the pressure to which a gas is subjected reduces its volume to one-half. Tripling the pressure, reduces the volume to one-third and so on.

[Pg 64]

Careful experiments reveal the following law: The volume of a given mass of gas at constant temperature is inversely proportional to the pressure to which it is subjected.

This law is often expressed mathematically. P/P´ = V´/V, or PV = P´V´. Since doubling the pressure reduces the volume one-half, it doubles the density. Tripling the pressure triples the density. We therefore have P/P´ = D/D´ or the density of a gas directly proportional to its pressure.

59. Height of the Atmosphere.—From its properties of compression and expansion, the air varies in density and pressure as one ascends in it. At a height of 3 miles the pressure is reduced to about one-half. This is an indication that one-half of the air is below this level. Balloonists have gone to a height of 7 miles, Glaser and[Pg 65] Coxwell in England in 1862 and Berson in France in 1901. The atmosphere has been explored to a height of 30,500 meters (18.95 miles) by sending up self-registering barometers in small balloons which burst at great altitudes. A parachute protects the instruments from breakage from too rapid fall. This height of 30,500 meters was reached by a balloon sent up by William R. Blair, at Huron, South Dakota, September 1, 1910.

At a height of 35 miles, the density is estimated at 1/30,000 of its value at sea-level. (See Fig. 38.) It is believed that some rarefied air exists for a considerable distance above this point, some estimates placing the extent at 100 miles, and others from 200 to 500 miles. Evidences of some air at such heights are shown by: (a) the height at which meteors first appear, (b) the height of the Aurora Borealis, and (c), the distance that the sun is below the horizon when the last traces of color disappear from the sky in the evening.

Although the exact limits of the atmosphere are unknown, the weight of a column of air 1 sq. cm. in cross-section, and extending upward as high as the atmosphere, may be accurately computed. For this column of air exactly balances the column of mercury in the tube of the barometer.

Below sea-level, the air increases rapidly in density and it is estimated that at a depth of 35 miles, the density of the air would be a thousand times that at the earth's surface, or more than that of water.

1. Evidence of compressibility of gases and incompressibility of liquids.

2. Boyle's Law. Proof, applications.

3. Extent of the atmosphere—three evidences.

[Pg 66]

Exercises

1. Mention three illustrations of the compressibility and expansibility of air that you know from your own experience.

2. Increasing the pressure increases the amount of a gas that will be absorbed by a liquid? Explain this. Have you ever observed this fact? Where?

3. If a toy balloon containing 2000 ccm. of gas at the earth's surface where the barometer reading is 76 cm., rises to an elevation where the barometer reads 54 cm., the balloon will tend to expand to what volume? Explain. Will it attain this volume?

4. If a gas is compressed, it changes in temperature. How do you explain this?

5. What change in temperature will occur when compressed air is allowed to expand? Explain.

6. Air blowing up a mountain side has its pressure lessened as it approaches the top. How will this affect the temperature? Why? What may result from this change in temperature? Explain.

7. To what pressure must 500 ccm. of air be subjected to compress it to 300 ccm. the barometer reading at first being 75 cm. Explain.

60. The Air Pump.—The air pump is used to remove air or other gases from a closed vessel. It was invented about 1650 by Otto Von Guericke, burgomaster of Magdeburg,[Pg 67] Germany. One form of air pump is shown in Fig. 39. C is a cylinder within which slides a tightly fitting piston. R is the vessel from which the air is to be exhausted. r and u are valves opening upward. The action of the pump is as follows:

On pushing the piston down, the air in C is compressed. This opens valve r allowing the confined air to escape above the piston. The piston is then raised making the space in C a partial vacuum. The pressure in R now being greater than in C, u is pushed up and the air from R rushes into C, until the pressure is equalized. On pushing down the piston again, valve u closes and the process is repeated until the pressure in R is no longer able to raise the valve u. Some air pumps are so constructed that the valves are opened and closed automatically by the movement of the piston. With these pumps a higher degree of rarefaction can be obtained.

Air is often partially exhausted from receivers or vessels by the use of a filter pump or aspirator. A stream of water flowing through a constriction causes a reduced pressure, draws in air and carries it away, and thus produces a partial vacuum. See Fig. 40 for a section of the device.

61. The Condensing Pump.—This is like the exhaust pump except that its valves are reversed. It is used in compressing illuminating gases into cylinders for use in lighting vehicles, stereopticons, Pintsch lights, gas light buoys, etc., and also for compressing air to operate air brakes, pneumatic hammers and drills, and for other uses.

The common condensing pump is the kind used for[Pg 68] inflating tires. (See Fig. 41.) In this, a loosely fitting metal piston is attached to a disc of leather somewhat larger than the cylinder. This device is called a cup valve. On raising the piston, air rushes in from the top past the valve, but on pushing the piston down, the valve is pressed tightly against the sides of the cylinder and prevents the escape of any air. The compressed air pushes open a valve on the tire and enters it. This valve closes as soon as the pressure is lessened from outside. It is well to notice in all of these pumps that two valves are used. One holds the air already secured while the other opens for a new supply. Both valves are never open at the same time.

62. Water Pumps.—The Common Lift Pump. This, the simplest pump for raising water, consists of a cylinder C (Fig. 42) connected by a pipe R to a supply of water as a cistern or well. A valve opening upward is placed at the bottom of the cylinder over the entrance to the pipe. In the cylinder is a tightly fitting piston connected by a rod to a lever for ease in action. The piston contains a valve opening upward. In operating this pump water is usually first poured into the cylinder to "prime" it. This helps to close the valves and prevents air leaking past them. When the piston is lowered the lower valve closes, the air in the cylinder being compressed pushes the upper valve open and passes above the piston. On raising the piston the upper valve closes. This forms a partial vacuum in the cylinder.

The air pressing on the surface of the water below forces the water and air that may be in the tube upward through the lower valve to fill this partial vacuum.

[Pg 69]

When the cylinder becomes filled with water, this is lifted out on the up-stroke, whence its name, "lift pump." Since the atmospheric pressure at sea-level can only support a column of water about 34 ft. high, the lower valve must be within this distance of the water surface. In actual practice the limit is about 27 ft. In deeper wells, the cylinder and valves are placed so that they are within 25 or 27 ft. of the surface of the water in the well, a long piston rod reaching above the surface of the ground and connected to a pump handle operates the piston. A discharge pipe extends from the cylinder to the surface of the ground above.

63. The Force Pump.—The force pump is used to deliver water under pressure either for spraying or to an elevated reservoir. The piston is solid, the second valve being placed at the entrance of the discharge pipe. (See Fig. 43.) The action is the same as that of the lift pump, with this exception; the piston in its down[Pg 70] stroke forces the water out through the discharge pipe, the velocity depending upon the pressure exerted.

A force pump is usually provided with an air chamber which is connected with the discharge pipe. On the down stroke of the piston, water is forced into the air chamber. This compresses the air it contains. The compressed air reacts and exerts pressure on the water forcing it out in a steady stream.

Force pumps are used in deep wells, being placed at the bottom.

The pumps used in city water works, fire engines, and all steam pumps, are force pumps. (See Fig. 44.)

64. The Siphon.—The siphon is a tube used to convey a liquid from one level over an elevation to a lower level by atmospheric pressure. It is used to remove liquids from tanks or vessels that have no opening at the bottom.

The siphon cannot be completely understood until one has mastered the laws of the flow of liquids. The following is offered as an incomplete explanation of its behavior. Consider the siphon to be full of water and[Pg 71] closed at d (Fig. 45). Atmospheric pressure on a will hold the siphon full if ab does not exceed 34 feet. If d is opened the water falls out with a speed equal to that acquired in falling from the level of a to that of d. This speed is acquired by all the water in the siphon and results in a drop in pressure throughout it. The pressure at a inside the siphon becomes less than the pressure at the same level outside as soon as the water starts flowing. The water in the vessel then flows into the siphon and out at d. This flow continues as long as there is a fall from the free surface of the water in the vessel to the outlet at d.

65. The Cartesian Diver.—This is a device which illustrates at the same time transmission of pressure by liquids, Archimedes' principle, and compressibility of gases. It was invented by Des Cartes (1596-1650). As ordinarily made, it is a hollow glass image with a small opening in the foot. It contains air and water in such amounts that the average density of image and contents is slightly less than that of water. It is placed in a tall glass jar filled with water and covered with tightly stretched rubber tissue. (See Fig. 46.) By pressing on the rubber cover the diver may be made to sink, since the air and water transmit the pressure on the cover which compresses the air inside the figure admitting some water to it, thus making the diver more[Pg 72] dense than water. By varying the pressure it can be made to sink, rise, or remain stationary at will.[D] A small vial can be used instead of the image.

66. Hydraulic Ram.—The hydraulic ram (see Fig. 47) is an automatic device that is much used for raising water from springs to houses located on higher ground. Water flows through the pipe A through the opening at B. The pressure closes the valve at B. The increased pressure in the pipe due to the closing of B opens the valve C and some of the water flows into the air chamber D. This reduces the pressure against the valve B so that it drops and allows a little water to escape. Just as this happens, valve C closes. The pressure in the pipe then closes B and forces water past C. This action being continually repeated, the air in D becomes so compressed that it has elastic force enough to raise the water in a steady stream to a height of many feet.

67. The Balloon.—Since air is a fluid, Archimedes' principle applies to it as well as to liquids. Therefore any object in the air is lifted up by a force equal to the weight of the air it displaces. The object will rise, if it weighs less than this displaced air and will continue to rise until both weights are equal.

The Balloon (Fig. 48) rises because it weighs less than the air it displaces, and therefore it is pushed up by the heavier air, the "lifting power" being the difference between its weight and that of the air displaced. The[Pg 73] neck at the bottom is left open to allow for expansion of the gas. When the aeronaut wishes to descend, he opens a valve at the top allowing some of the gas to escape.

Hydrogen is the lightest gas, weighing 0.09 kg. per cubic meter, and so gives the greatest lifting power, but as it is expensive to make, coal gas, density 0.75 kg. per cubic meter, is ordinarily employed. Helium has recently been used to fill military balloons because it cannot be set on fire.

[Pg 74]

The Parachute (Fig. 49) is an umbrella-shaped device for use in descending from a balloon. After falling a few seconds it opens, the large surface exposed to the air causing it to descend slowly. The hole in the top keeps the parachute upright by allowing the air to escape through it, thus relieving the pressure.

68. The Air Brake.—Compressed air is used to do work in many machines, such as pneumatic drills, hammers, and air brakes. The Westinghouse air brake (Fig. 50) uses air at a pressure of about 70 lbs. to the square inch. The essential parts as shown are a reservoir R, the brake cylinder C and a triple valve V, placed under each car with an air pipe P, leading to the engine. This is connected to R by the triple valve V. When the pressure in P is reduced by the engineer or by accident, the triple valve operates so as to admit air from R into the cylinder C pushing the piston H to the left. H is connected to the brakes by levers which press the brake shoes strongly against the wheels. When the air pressure in P is restored the triple valve acts so as to permit the air in C to escape while R is filled again from P. The hissing sound heard when a train stops is caused by air escaping from cylinder C. The[Pg 75] spring in C keeps the brakes from the wheels except when the "air is on."

69. The Gas Meter.—The gas meter consists of a box divided into two parts by a vertical partition (Fig. 51). Two bellows are attached to this partition, one on each side. The valves that regulate the flow of gas to and from the bellows and the chambers A and D are opened and closed by levers connected with the bellows. These levers also operate the hands upon the dials. When the inlet to the bellows B is opened, the outlet of A is also opened. Gas entering B opens the bellows and forces the gas in A out into the house-pipe E. When B is full its inlet valve closes and its outlet valve opens. The inlet of A also opens and its outlet closes. Gas now flows into A, compressing the bellows and B, and forcing the gas from it into the house-pipe. At each filling of the bellows B there will be displaced from A and forced into the house-pipe as much gas as enters B. It is evident that at each emptying of B an equal amount of gas enters A. Thus we have A and B alternately filling and emptying as long as the gas burner is open. To have a continuous flow of gas in the house-pipes two pipes and two chambers are[Pg 76] necessary, one being filled while the other is being emptied.

Fig. 52 represents the dials upon a gas meter showing a reading of 54,600 cu. ft.

70. Centrifugal Pumps. Fluids, such as water and air, are often put in motion by devices called centrifugal pumps (see Art. 78). These pumps contain a revolving part, like a wheel without a rim, whose spokes are replaced by thin blades. This revolving part resembles the paddle wheel of some steam boats and is enclosed in a case or cover having one opening at the rim and another opening on one side about the axle.

When the wheel is rapidly revolved, the fluid is driven out with considerable force through the opening at the rim, while a partial vacuum is produced at the axle causing a rapid flow into the device at this point.

This is the principle of the action of the vacuum cleaner.[Pg 77] Fig. 53 is a section of a vacuum sweeper showing the revolving wheel and the current of air passing into the wheel at the lower side and out of the rim of the case at the rear.

Centrifugal water pumps work on the same principle and furnish a continuous flow of water, often large in volume and at considerable pressure.

1. Air pump.

2. Condensing pump.

3. Lift and force pumps.

4. Siphon.

5. Cartesian diver.

6. Hydraulic ram.

7. Balloon.

8. Air brake.

9. Gas meter.

10. Vacuum cleaner.

1. Explain why smoke settles to the ground before storms.

2. Why does the water rise in the suction pipe of a pump?

3. Why is it easier to float in water when the lungs are filled with air than when they are not filled?

4. Why is it easier to swim in salt water than in fresh water?

5. How are submarines made to sink? to rise to the surface?

6. How can a fish rise or sink in water?

7. Explain why a life preserver made of cork will enable a person to float.

8. Hold the open hand out flat with the fingers together. Place underneath the fingers a piece of paper. Blow between the first and second fingers against the paper. As long as you blow hard the paper will not fall but will stick to the hand. Explain.

9. Why does pressing the bulb of an atomizer force out the liquid in a fine spray?

10. Why is air that contains a large amount of water vapor lighter than air that only contains a small amount?

11. How are heights above sea-level ascertained by a barometer?

12. Oil floats on water but sinks in alcohol. Explain.

13. In a balloon the lower end is often open to the air. Why does not the gas escape and prevent the balloon from rising?

14. How long will a balloon continue to rise?

[Pg 78]

15. If the pressure against the 8-in. piston of an air brake is 70 lbs. per square inch, how much force does the piston exert?

16. The capacity of a balloon is 40,000 cu. ft. The weight of the balloon, car, etc., is 600 lbs.; specific gravity of the gas used is 0.46 that of the air. Find how much weight the balloon can carry.

17. The so-called Magdeburg hemispheres were invented by Otto von Guericke of Magdeburg, Germany. When the hemispheres (see Fig. 54) are placed in contact and the air exhausted it is found very difficult to pull them apart. Explain.

18. Von Guericke's hemispheres had an inside diameter of 22 in. What force would be required to pull them apart if all the air were exhausted from them? (Find the atmospheric force on a circle, 22 in. in diameter.)

19. Von Guericke made a water barometer whose top extended through the roof of his house. On the top of the water in the tube was placed a wooden image. In fair weather the image appeared above the roof, but it descended before a storm. Explain.

20. The balloon "Goodyear" (Fig. 48), which won the International championship race at Paris in 1913, has a capacity of 80,000 cu. ft. The gas bag weighs 653 lbs., the net 240 lbs. and the basket 92 lbs. How large a load can it carry when filled with hydrogen specific gravity 0.069 (compared with air).

Liquids: Force, pressure, and density. Floating and immersed bodies. Laws: Liquid force, F = A.h.d, Pascal's, Archimedes. Illustrations and Applications:

Devices: Hydraulic press, air cushion, barometer—mercurial and aneroid. Pumps, lift, force, vacuum, compression, centrifugal, balloon, siphon, etc. Construction and action of each.

[Pg 79]

FORCE AND MOTION

71. Force.—We have been studying various forces, such as air pressure, pressure in liquids, and the force of elasticity in solids, and have considered them simply as pushes or pulls. A more formal study of forces in general and of devices for representing and measuring them will be helpful at this point of the course.

A force is that which tends to cause a change in the size or shape of a body or in its state of motion. In other words a force is a push or a pull. That is, force tends to produce distortion or change of motion in a body. Force itself is invisible. We measure it by the effect it produces. Forces are usually associated with the objects exerting them. Thus we speak of muscular force, air pressure, liquid pressure, the force of a spring, the force of the earth's attraction and so on.

Forces are classified in various ways.

I. With respect to the duration and steadiness of the force.

(a) Constant, as the earth's attraction. (b) Impulsive, as the stroke of a bat on a ball. (c) Variable, as the force of the wind.

II. With respect to the direction of the force.

(a) Attractive, as the earth's attraction. (b) Repulsive, as air pressure, liquid pressure, etc.

72. Methods of Measuring Force.—Since forces are measured by their effects which are either distortion or change of motion, either of these effects may be used to[Pg 80] measure them. For example, the force exerted by a locomotive is sometimes computed by the speed it can develop in a train of cars in a given time, or the force of the blow of a baseball bat is estimated by the distance the ball goes before it strikes the ground.

The more common method of measuring force, however, is by distortion, that is, by measuring the change of shape of a body caused by the force. In doing this, use is made of Hooke's Law (Art. 32), in which it is stated that "within the limits of perfect elasticity," changes of size or shape are directly proportional to the forces employed. That is, twice as great a force will produce twice as great a change of shape and so on.

A common contrivance using this principle is the spring balance (Fig. 55), with which all are familiar, as ice scales, meat scales, postal scales, etc. The object which changes shape in this device is a coiled spring contained in the case of the instrument. The balance is so constructed that when the spring is pulled out as far as possible it has not reached its limit of elasticity, since, if the spring were stretched so as to exceed its elastic limit, the index would not return to its first position on removing the load. (See Arts. 30-32.)

73. Graphic Representation of Forces.—A force is said to have three elements. These are (a) its point of application, (b) its direction, and (c) its magnitude. For example, if there is hung upon the hook of a spring balance a weight of 5 lbs., then we have: (a) its point of application on the hook of the balance, (b) its downward direction[Pg 81] and (c) its magnitude, or 5 lbs. These three elements may be represented by a line. Thus in Fig. 56a, a line AB is drawn as shown, five units long; A represents the point of application; B, the arrow head, shows the direction; and the length of the line (five units) shows the magnitude of the force.

This is called a graphic representation since it represents by a line the quantity in question. If another weight of 5 lbs. were hung from the first one, the graphic representation of both forces would be as in Fig. 56b. Here the first force is represented by AB as before, BC representing the second force applied. The whole line represents the resultant of the two forces or the result of their combination. If the two weights were hung one at each end of a short stick AC (Fig. 56c), and the latter suspended at its center their combined weight or resultant would of course be applied at the center. The direction would be the same as that of the two weights. The resultant therefore is represented by ON. In order to exactly balance this resultant ON, a force of equal magnitude but opposite in direction must be applied at the point of application of ON, or O. OM then represents a force that will just balance or hold in equilibrium the resultant of the two forces AB and CD. This line OM therefore represents the equilibrant of the weights AB and CD. The resultant of two forces at an angle with each other is formed differently,[Pg 82] as in Fig. 57 a. Here two forces AB and AC act at an angle with each other. Lay off at the designated angle the lines AB and AC of such length as will accurately represent the forces. Lay off BD equal to AC and CD equal to AB. The figure ABCD is then a parallelogram. Its diagonal AD represents the resultant of the forces AB and AC acting at the angle BAC. If BAC equals 90 degrees or is a right angle, AD may be computed thus: AB² + BD² = AD². Why?

and AD = √([line]AB² + [line]BD²).

This method of determining the resultant by computation may be used when the two forces are at right angles. (In any case, AD may be measured using the same scale that is laid off upon AB and AC, as shown in Fig. 57 b.) The three cases of combining forces just given may be classified as follows: The first is that of two forces acting along the same line in the same or opposite direction, as when two horses are hitched tandem, or in a tug of war. The second is that of two forces acting along parallel lines, in the same direction, as when two horses are hitched side by side or abreast. The third is that of two forces acting at the same point at an angle. It may be represented by the device shown in Fig. 58, consisting of two spring balances[Pg 83] suspended from nails at the top of the blackboard at A and B. A cord is attached to both hooks and is passed through a small ring at O from which is suspended a known weight, W. Lines are drawn on the blackboard under the stretched cords, from O toward OA, OB, and OW and distances measured on each from O to correspond to the three forces as read on balance A and B and the weight W. Let a parallelogram be constructed on the lines measured off on OA and OB. Its diagonal drawn from O will be found to be vertical and of the same length as the line measured on OW. The diagonal is the resultant of the two forces and OW is the equilibrant which is equal and opposite to the resultant.

Again, the first case may be represented by a boat moving up or down a stream; the resultant motion being the combined effect of the boat's motion and that of the stream. The second, may be represented by two horses attached side by side to the same evener. The resultant force equals the sum of the two component forces. The third, may be represented by a boat going across a stream, the resultant motion being represented by the diagonal of the parallelogram formed by using the lines that represent the motion of the stream and of the boat.

74. Units for Measuring Force.—Force is commonly measured in units of weight: in pounds, kilograms, and grams. For example, we speak of 15 lbs. pressure per[Pg 84] square inch and 1033.6 g. pressure per square centimeter as representing the air pressure. It should be noted here that the words pound, kilogram, and gram are used not only to represent weight or force but also the masses of the objects considered. Thus, one may speak of a pound-mass meaning the amount of material in the object.

It will help to avoid confusion if we reserve the simple terms "gram" and "pound" to denote exclusively an amount of matter, that is, a mass, and to use the full expression "gram of force" or "pound of force" whenever we have in mind the pull of the earth upon these masses. Or, one may speak of a pound-weight meaning the amount of attraction exerted by the earth upon the object. The same is true of gram-mass and gram-weight. The mass of a body does not change when the body is transferred to another place. The weight, however, may vary, for on moving a body from the equator toward the poles of the earth the weight is known to increase.

1. Definition of force.

2. Classification of forces. (a) Duration: constant, impulsive, variable. (b) Direction: attractive, repulsive.

3. Methods of measuring force. (a) By distortion. (b) By change of motion.

4. Graphic representation of forces: component, resultant, equilibrant.

5. Three cases of combining forces. (1) Two forces acting on the same line. (2) Two forces acting in parallel lines. (3) Two forces acting at the same point at an angle.

6. Units for measuring force, pound, gram.

1. Name five natural forces. Which produce a tension? Which a pressure?

2. How much can you lift? Express in pounds and kilograms.

[Pg 85]

3. Show graphically the resultant of two forces at right angles, one of 12 lbs., the other of 16 lbs. What is the magnitude of this resultant? Then determine the answer, first by measurement and then by computation. Which answer is more accurate? Why?

4. Represent by a parallelogram the two forces that support a person sitting in a hammock and draw the line representing the resultant.

5. Find graphically the resultant of the pull of two forces, one of 500 lbs. east and one of 600 lbs. northwest.

6. Determine the equilibrant of two forces, one of 800 lbs. south and one of 600 lbs. west.

7. Would the fact that weight varies on going from the equator to either pole be shown by a spring balance or a beam balance? Explain.

75. Motion a Change of Position.—Motion is defined as a continuous change in the position of a body. The position of a body is usually described as its distance and direction from some fixed point. Thus a man on a boat may be at rest with respect to the boat and moving with respect to the earth. Or, if he walks toward the stern as fast as the boat moves forward, he may keep directly over a rock on the bottom of the lake and hence not be moving with reference to the rock and yet be in motion with respect to the boat. Motion and rest, therefore, are relative terms. The earth itself is in motion in turning on its axis, in moving along its orbit, and in following the sun in its motion through space. Motions are classified in several ways:

1. Translation.—A body is said to have motion of translation when every line in it keeps the same direction.

2. Rotation.—A body has motion of rotation when it[Pg 86] turns upon a fixed axis within the body, as a wheel upon its axle or the earth upon its axis.

3. Vibration or Oscillation.—A body is said to have vibratory or oscillatory motion when it returns to the same point at regular intervals by reversals of motion along a given path, e.g., a pendulum.

1. Rectilinear.—A body has rectilinear motion when its path is a straight line. Absolute rectilinear motion does not exist, although the motion of a train on a straight stretch of track is nearly rectilinear.

2. Curvilinear.—A body has curvilinear motion when its path is a curved line, e.g., the path of a thrown ball.

1. Uniform.—A body has uniform motion when its speed and direction of motion do not change. Uniform motion for extended periods is rarely observed. A train may cover, on an average, 40 miles per hour but during each hour its speed may rise and fall.

2. Variable.—A body has variable motion when its speed or direction of motion is continually changing. Most bodies have variable motion.

3. Accelerated.—A body has accelerated motion when its speed or direction of motion continually changes. If the speed changes by the same amount each second, and the direction of motion does not change the motion is said to be uniformly accelerated, e.g., a falling body.

Uniformly accelerated motion will be studied further under the topic of falling bodies.

Velocity is the rate of motion of a body in a given direction. For example, a bullet may have a velocity of 1300 ft. a[Pg 87] second upwards. Acceleration is the rate of change of velocity in a given direction, or the change of velocity in a unit of time. A train starting from a station gradually increases its speed. The gain in velocity during one second is its acceleration. When the velocity is decreasing, as when a train is slowing down, the acceleration is opposite in direction to the velocity. A falling body falls faster and faster. It has downward acceleration. A ball thrown upward goes more and more slowly. It also has downward acceleration.

76. Momentum.—It is a matter of common observation that a heavy body is set in motion with more difficulty than a light one, or if the same force is used for the same length of time upon a light and a heavy body,[E] the light body will be given a greater velocity. This observation has led to the calculation of what is called the "quantity of motion" of a body, or its momentum. It is computed by multiplying the mass by the velocity. If the C.G.S. system is used we shall have as the momentum of a 12 g. body moving 25 cm. a second a momentum of 12 × 25 or 300 C.G.S. units of momentum. This unit has no name and is therefore expressed as indicated above. The formula for computing momentum is: M = mv.

77. Inertia, First Law of Motion.—One often observes when riding in a train that if the train moves forward suddenly the passengers do not get into motion as soon as the train, and apparently are jerked backward. While if the train is stopped suddenly, the passengers tend to keep in motion. This tendency of matter to keep moving when in motion and to remain at rest when at rest is[Pg 88] often referred to as the property of inertia. Newton's first law of motion, often called the law of inertia, describes this property of matter as follows:

Every body continues in a state of rest or of uniform motion in a straight line unless it is compelled to change that state by some external force. This means that if an object like a book is lying on a table it will remain there until removed by some outside force. No inanimate object can move itself or stop itself. If a ball is thrown into the air it would move on forever if it were not for the force of attraction of the earth and the resistance of the air.

It takes time to put a mass into motion, a heavy object requiring more time for a change than a light object. As an example of this, note the movements of passengers in a street car when it starts or stops suddenly. Another illustration of the law of inertia is the so-called "penny and card" experiment. Balance a card on the end of a finger. Place on it a coin directly over the finger, snap the card quickly so as to drive the card from beneath the coin. The coin will remain on the finger. (See Fig. 59.)

According to Newton's first law of motion a moving body which could be entirely freed from the action of all external forces would have uniform motion, and would describe a perfectly straight course. The curved path taken by a baseball when thrown shows that it is acted upon by an outside force. This force, the attraction of the earth, is called gravity.

78. Curvilinear Motion.—Curvilinear motion occurs when a moving body is pulled or pushed away from a[Pg 89]
[Pg 90]
[Pg 91] straight path. The pull or push is called centripetal (center-seeking) force. A moving stone on the end of a string when pulled toward the hand moves in a curve. If the string is released the stone moves in a tangent to the curve. The string pulls the hand. This phase of the pull is called centrifugal force. The centripetal force is the pull on the stone. Centripetal and centrifugal force[Pg 92] together cause a tension in the string. Examples of curvilinear motion are very common. The rider and horse in a circus ring lean inward in order to move in a curve. The curve on a running track in a gymnasium is "banked" for the same reason. Mud flying from the wheel of a carriage, the skidding of an automobile when passing rapidly around a corner, and sparks flying from an emery wheel, are illustrations of the First Law of Motion.

Cream is separated from milk by placing the whole milk in a rapidly revolving bowl, the cream being lighter collects in the center and is thrown off at the top. (See Fig. 60.) Clothes in steam laundries are dried by a centrifugal drier. In amusement parks many devices use this principle. (See centrifugal pumps, Art. 70.)

79. The Second Law of Motion, sometimes called the law of momentum, leads to the measurement of force, by the momentum or the quantity of motion, produced by it. The law is stated as follows:

Change of motion, or momentum, is proportional to the acting force and takes place in the direction in which the force acts. In other words, if two or more forces act at the same instant upon a body each produces the same effect that it would if acting alone. If a card be supported on two nails driven horizontally close together into an upright board (see Fig. 61), and two marbles be so placed on the ends as to balance each other, when one marble is snapped horizontally by a blow, the other will fall. Both reach the floor at the same time. The two balls are equally pulled down by the earth's attraction and strike the ground at the same time, though one is shot sidewise, and the other is dropped vertically.

[Pg 93]

As gravity is a constant force, while the blow was only a momentary force, the actual path or resultant motion will be a curved line.

The constant relation, between the acting force and the change of momentum it produces in a body, has led to the adoption of a convenient C.G.S. unit of force called the dyne. The dyne is that force which can impart to a mass of one gram a change of velocity at the rate of one centimeter per second every second. This definition assumes that the body acted upon is free to move without hindrance of any kind, so that the acting force has to overcome only the inertia of the body. However, the law applies in every case of application of force, so that each force produces its full effect independently of other forces that may be acting at the same time upon the body.

80. Newton's Third Law.—This law has been experienced by everyone who has jumped from a rowboat near the shore. The muscular action that pushes the body forward from the boat also pushes the boat backward, often with awkward results. The law is stated: To every action, there is always an opposite and equal reaction, or the mutual actions of any two bodies are always equal and opposite in direction. Many illustrations of this law are in every one's mind: a stretched rope pulls with the same force in one direction as it does in the opposite direction. If a bat hits a ball, the ball hits the bat with an equal and opposite force. The third law is therefore sometimes called the law of reaction. When a weight is hung upon a spring balance the action of the weight pulls down the spring until it has stretched sufficiently (Hooke's Law) to produce an elastic reaction that equals and hence supports the weight. When a man stands at the center of a plank supported at its ends, the action of the man's weight bends the plank until the elastic force developed[Pg 94] in the plank equals the weight applied. Further, when a train or a wagon is on a bridge the bridge yields until it has developed an elastic reaction equal to the weight applied. If a person stands in the center of a room, the floor beams yield until the third law is satisfied. In fact, whenever a force acts, a contrary equal force always acts.

81. Stress and Strain.—A pair of forces that constitute an action and a reaction is called a stress. The two forces are two parts of one stress. If the two forces act away from each other, as in the breaking of a string, the stress is called a tension, but if they act toward each other as in crushing anything, the stress is called a pressure. In order for a body to exert force it must meet with resistance. The force exerted is never greater than the resistance encountered. Thus one can exert but little force upon a feather floating in the air or upon other light objects. A fast moving shot exerts no force unless it encounters some resistance.

Forces, then, are always found in pairs. Thus to break a string, to stretch an elastic band, to squeeze a lemon, one must exert two equal and opposite forces. Such a thing as a single force acting alone is unknown. Usually, however, we give our attention mainly to one of the forces and ignore the other. When a force acts upon a body the change of shape or size resulting is called a strain. Hooke's law (Art. 32) is often expressed as follows: "The strain is proportional to the stress," e.g., the stretch of the spring of a spring balance is proportional to the load placed upon it.

1. Motion a change of position. Kinds of motion.

2. Newton's laws of motion.

3. Momentum.

4. Inertia. First law of motion. Curvilinear motion.

[Pg 95]

5. Second law of motion.

6. Third law of motion. Action and reaction, stress and strain.

1. Mention three illustrations of the third law, different from those given.

2. A rifle bullet thrown against a board standing upon edge will knock it down; the same bullet fired at the board will pass through it without disturbing its position. Explain.

3. A hammer is often driven on to its handle by striking the end of the latter. Explain.

4. Consider a train moving 60 miles an hour, with a gun on the rear platform pointing straight backward. If a ball is fired from the gun with a speed of 60 miles an hour, what will happen to the ball?

5. Could one play ball on the deck of an ocean steamer going 25 miles an hour without making allowance for the motion of the ship? Explain.

6. On a railroad curve, one rail is always higher. Which? Why?

7. Why can a small boy when chased by a big boy often escape by dodging?

8. Will a stone dropped from a moving train fall in a straight line? Explain.

9. A blast of fine sand driven against a sheet of glass soon gives it a rough surface. Explain.

10. Explain the use of fly-wheels in steadying the motion of machinery (for example, the sewing machine).

11. Is it easier to walk to the front or rear of a passenger train when it is stopping? Why?

12. Why does lowering the handles of a wheel-barrow on the instant of striking make it easier to go over a bump?

13. Why should a strong side wind interfere with a game of tennis? How can it be allowed for?

14. On which side of a railroad track at a curve is it the safer to walk while a train is passing? Why?

15. Why does a bullet when fired through a window make a clean round hole in the glass, while a small stone thrown against the window shatters the glass?

16. A tallow candle can be fired through a pine board. Why?

[Pg 96]

17. In cyclones, straws are frequently found driven a little distance into trees; why are the straws not broken and crushed instead of being driven into the tree unbroken?

18. A bullet weighing one-half oz. is fired from a gun weighing 8 lb. The bullet has a velocity of 1800 ft. per second. Find the velocity of the "kick" or recoil of the gun.

18. When football players run into each other which one is thrown the harder? Why?

20. A railroad train weighing 400 tons has a velocity of 60 miles per hour. An ocean steamer weighing 20,000 tons has a velocity of one half mile per hour. How do their momenta compare?

21. Why is a heavy boy preferable to a lighter weight boy for a football team?

22. Why does a blacksmith when he desires to strike a heavy blow, select a heavy sledge hammer and swing it over his head?

23. Why does the catcher on a baseball team wear a padded glove?

82. Resolution of Forces.—We have been studying the effect of forces in producing motion and the results of combining forces in many ways; in the same line, in parallel lines, and in diverging lines. Another case of much interest and importance is the determination of the effectiveness of a force in a direction different from the one in which it acts. This case which is called resolution of forces is frequently used. To illustrate: one needs but to recall that a sailor uses this principle in a practical way whenever he sails his boat in any other direction than the one in which the wind is blowing, e.g., when the wind is blowing, say from the north, the boat may be driven east, west, or to any point south between the east and west and it is even possible to beat back against the wind toward the northeast or northwest. Take a sled drawn by a short rope with the force applied along the line AB (see Fig. 62); part of this force tends to lift the front of the sled as AC and a part to draw it forward as AD. Hence not all of the force applied along[Pg 97] AB is used in drawing the sled forward. Its effectiveness is indicated by the relative size of the component AD compared to AB.

The force of gravity acting upon a sphere that is resting on an inclined plane may be readily resolved into two components, one, the effective component, as OR, and the other, the non-effective as OS. (See Fig. 63.) If the angle ACB is 30 degrees, AB equals 1/2 of AC and OR equals 1/2 of OG, so that the speed of the sphere down the plane developed in 1 second is less than (about one-half of) the speed of a freely falling body developed in the same time. Why is OS non-effective?

83. The Aeroplane.—The aeroplane consists of one or two frames ABCD (see Fig. 64), over which is stretched cloth or thin sheet metal. It is driven through the air by a propeller turned by a powerful gasoline motor. This has the effect of creating a strong breeze coming toward the front of the aeroplane. As in the case of the sailboat a pressure is created at right angles to the plane along GF and this may be resolved into two components[Pg 98] as GC and GE, GC acting to lift the aeroplane vertically and GE opposing the action of the propeller. Fig. 65 represents the Curtis Flying Boat passing over the Detroit river.

1. If a wagon weighing 4000 lbs. is upon a hill which rises 1 ft. in 6, what force parallel to the hill will just support the load? (Find the effective component of the weight down the hill.)

2. If a barrel is being rolled up a 16-ft. ladder into a wagon box 3 ft. from the ground, what force will hold the barrel in place on the ladder, if the barrel weighs 240 lbs. Show by diagram.

3. Show graphically the components into which a man's push upon the handle of a lawn mower is resolved.

4. Does a man shooting a flying duck aim at the bird? Explain.

5. What are the three forces that act on a kite when it is "standing" in the air?

6. What relation does the resultant of any two of the forces in problem five have to the third?

[Pg 99]

7. Into what two forces is the weight of a wagon descending a hill resolved? Explain by use of a diagram.

8. A wind strikes the sail of a boat at an angle of 60 degrees to the perpendicular with a pressure of 3 lbs. per square foot. What is the effective pressure, perpendicular to the sail? What would be the effective pressure when it strikes at 30 degrees?

9. How is the vertical component of the force acting on an aeroplane affected when the front edge of the plane is elevated? Show by diagram.

84. Moment of Force.—In the study of motion we found that the quantity of motion is called momentum and is measured by the product of the mass times the velocity. In the study of parallel forces, especially such as tend to produce rotation, we consider a similar quantity. It is called a moment of force, which is the term applied to the effectiveness of a force in producing change of rotation. It also measured by the product of two quantities; One, the magnitude of the force itself, and the other, the perpendicular distance from the axis about which the rotation takes place to the line representing the direction of the force.

To illustrate: Take a rod, as a meter stick, drill a hole at S and place through it a screw fastened at the top of the blackboard. Attach by cords two spring balances and draw to the right and left, A and B as in Fig. 66. Draw out the balance B about half way, hold it steadily, or fasten the cord at the side of the blackboard, and read both balances. Note also the distance AS and BS. Since the rod is at rest, the tendency to rotate to the right and left must be equal. That is, the moments of the forces[Pg 100] at A and B about S are equal. Since these are computed by the product of the force times the force arm, multiply B by BS and A by AS and see if the computed moments are equal. Hence a force that tends to turn or rotate a body to the right can be balanced by another of equal moment that acts toward the left.

85. Parallel Forces.—Objects are frequently supported by two or more upward forces acting at different points and forming in this way a system of parallel forces; as when two boys carry a string of fish on a rod between them or when a bridge is supported at its ends. The principle of moments just described aids in determining the magnitude of such forces and of their resultant. To illustrate this take a wooden board 4 in. wide and 4 ft. long of uniform dimensions. (See Fig. 67.) Place several screw hooks on one edge with one set at O where the board will hang horizontally when the board is suspended there. Weigh the board by a spring balance hung at O. This will be the resultant in the following tests. Now hang the board from two spring balances at M and N and read both balances. Call readings f and f´. To test the forces consider M as a fixed point (see Fig. 67) and the weight of the board to act at O. Then the moment of the weight of the board should be equal the moment of the force at N since the board does not move, or w times OM equals f´ times NM. If N is considered the fixed point then the moment of the weight of the board and of f with reference to the point N should be equal, or w times ON = f times NM. Keeping this illustration in mind, the law of parallel forces may be stated at follows: 1. The resultant of two parallel forces acting in the same direction at different points in a body is equal to their sum and has the same direction as the components.

[Pg 101]

The moment of one of the components about the point of application of the other is equal and opposite to the moment of the supported weight about the other.

86. The Couple.—If two equal parallel forces act upon a body along different lines in opposite directions, as in Fig. 68, they have no single resultant or there is no one force that will have the same effect as the two components acting together. A combination of forces of this kind is called a couple. Its tendency is to produce change of rotation in a body. An example is the action upon a compass needle which is rotated by a force which urges one end toward the north and by an equal force which urges the other end toward the south.

1. Moment of force, how measured.

2. Parallel forces.

3. The two laws of parallel forces.

4. The couple.

[Pg 102]

1. Show by diagram how to arrange a three-horse evener so that each horse must take one-third of the load.

2. Two boys support a 10-ft. pole on their shoulders with a 40-lb. string of fish supported from it 4 ft. from the front boy. What load does each boy carry? Work by principle of moments.

3. If two horses draw a load exerting a combined pull of 300 lbs., what force must each exert if one is 28 in. and the other is 32 in. from the point of attachment of the evener to the load?

4. A weight of 100 lbs. is suspended at the middle of a rope ACB 20 ft. long. (See Fig. 69.) The ends of the rope are fastened at points A and B at the same height. Consider D as the center of the line AB. What is the tension of the rope when CD is 3 ft.? When CD is 1 ft.? When CD is 1 in.?

5. A crane is set up with the tie horizontal. (See Fig. 70.) If 1000 lbs. is to be lifted, find the tie stress and the boom stress if the boom angle is 30 degrees? If 45 degrees? 60 degrees?

6. A ball is placed on a plane inclined at an angle of 30 degrees to the horizontal. What fraction of its weight tends to cause motion down the plane? What effect does the other component of the weight have? Why?

7. A person weighing 150 lbs. is lying in a hammock. The distance between the supports is 15 ft. The hammock sags 4 ft. What is the tension in the supports at each end? What is the tension when the sag is only 1 ft.?

[Pg 103]

8. A ladder 30 ft. long and weighing 80 lbs. leans against the side of a building so that it makes an angle of 30 degrees with the building. Find the direction and magnitude of the component forces on the ground and at the building.

9. A traveling crane 50 ft. long weighing 10 tons moves from one end of a shop to the other, at the same time a load of 4000 lbs. moves from end to end of the crane. Find the pressure of the trucks of the crane on the track when the load is at a distance of 5, 10, 15, and 25 ft. from either end.

10. Resolve a force of 500 lbs. into two components at right angles to each other, one of which shall be four times the other.

11. A truss (see Fig. 71), carries a load of 1000 lbs. at C. Find the forces acting along AC, BC, and AB. If AC and BC are each 12 ft. and AB 20 ft., which of these forces are tensions and which are pressures?

87. Gravitation.—Gravitation is the force of attraction that exists between all bodies of matter at all distances. This attraction exists not only between the heavenly bodies, the stars and planets, etc., but is also found between bodies on the earth. A book attracts all objects in a room and outside of a room as well, since its weight shows that it is attracted by the earth itself. The gravitational attraction between ordinary bodies is so slight that it requires careful experiments to detect it. In fact, it is only when one of the attracting bodies is large, as for example the earth, that the force becomes considerable. Careful studies of the motions of the heavenly bodies, especially of that of the moon in its orbit about the earth, led Sir Isaac Newton to the statement of the law of gravitation which is well expressed in the following statement:

[Pg 104]

88. Law of Gravitation.—Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

The law may be separated into two parts, one referring to the masses of the bodies concerned, the other to the effect of the distance between them. The first part is easily understood since we all know that two quarts of milk will weigh just twice as much as one quart. To illustrate the second part of the law, suppose that the moon were removed to twice its present distance from the earth, then the attraction between the earth and the moon would be one-fourth its present attraction. If removed to three times its present distance, the attraction would be one-ninth, etc.

The attraction of the earth for other bodies on or near it is called gravity. The weight of a body is the measure of the earth's attraction for it; or it is the force of gravity acting upon it. Newton's third law of motion states that every action is accompanied by an equal and opposite reaction (Art. 80). Hence, the attraction of the earth for a book or any other object is accompanied by an equal attraction of the book for the earth.

89. Weight.—In advanced physics it is proved that a sphere attracts as if it were concentrated at its center. Thus if the earth's radius be considered as 4000 miles, then a body 4000 miles above the earth's surface would be 8000 miles above the earth's center, or twice as far from the center of the earth as is a body upon the earth's surface. A body then 4000 miles above the earth's surface will weigh then but one-fourth as much as it will at the surface of the earth.

Since the earth is flattened at the poles, the surface at[Pg 105] the equator is farther from the center of the earth than at points north or south. Thus a body weighing 1 lb. at the equator weighs 1.002 lb. at Chicago, or about 1/500 more. The rotation of the earth also affects the weight of a body upon it so that at the equator the weight of a body is 1/289 less than at the pole. Both effects, that of flattening and of rotation, tend to diminish the weight of bodies at the equator, so that a body at the latter place weighs about 1/192 less than at the poles.

90. Center of Gravity.—A body is composed of a great many particles each of which is pulled toward the center of the earth by the force of gravity. A single force that would exactly equal the combined effect of the pull of the earth for all the particles of a body would be their resultant. The magnitude of this resultant is the weight of the body. The direction of this resultant is in a line passing toward the earth's center, while the point of application of this[Pg 106] resultant is called the center of gravity of the body. The center of gravity of a body may also be briefly defined as the point about which it may be balanced. As the location of this point depends upon the distribution of matter in the body, the center of gravity is also sometimes called the center of mass of the body.

The earth's attraction for a body is considered for the sake of simplicity, not as a multitude of little forces, but as a single force applied at its center of gravity. To find the center of gravity of a body find two intersecting lines along which it balances, see Fig. 72, and the center of gravity will be at the intersection. A vertical line through this point is sometimes called the line of direction of the weight.

91. Equilibrium of Bodies.—Equilibrium means equally balanced. A body at rest or in uniform motion is then in equilibrium. An object is in equilibrium under gravity when a vertical line through its center of gravity passes through the point of support. A trunk is an example of a body in equilibrium since a vertical line from its center of gravity falls within the base formed by the area upon which it rests. Work will be necessary to tip the trunk from its position. The amount of work required will depend upon the weight of the body and the location of the center of gravity.

92. Kinds of Equilibrium.—(a) Stable.—A body is in stable equilibrium under gravity if its center of gravity is raised whenever the body is displaced. It will return to its first position if allowed to fall after being slightly displaced. In Fig. 73, a and b if slightly tipped will return to their first position. They are in stable equilibrium.[Pg 107] Other examples are a rocking chair, and the combination shown in Fig. 74.

(b) Unstable.—A body is in unstable equilibrium under gravity if its center of gravity is lowered whenever the body is slightly displaced. It will fall farther from its first position. A pencil balanced on its point or a broom balanced on the end of the handle are in unstable equilibrium. The slightest disturbance will make the line of direction of the weight fall outside of (away from) the point of support (Fig. 75 a).

(c) Neutral.—A body is in neutral equilibrium if its center of gravity is neither raised nor lowered whenever the body is moved. Familiar examples are a ball lying on a table (Fig. 75 b) and a wagon moving on a level street (referring to its forward motion).

[Pg 108]

93. Stability.—When a body is in stable equilibrium, effort must be exerted to overturn it, and the degree of stability is measured by the effort required to overturn it. To overturn a body, it must be moved so that the vertical line through its center of gravity will pass outside of its supporting base. This movement in stable bodies necessitates a raising of the center of gravity. The higher this center of gravity must be raised in overturning the body, the more stable it is, e.g., see Fig. 76. Thus a wagon on a hillside will not overturn until its weight falls outside of its base, as in Fig. 77 B. The stability of a body depends upon the position of its center of gravity and the area of its base. The lower the center of gravity and the larger the base, the more stable the body. What means are employed to give stability to bodies, in every-day use (such as clocks, ink-stands, pitchers, vases, chairs, lamps, etc.)?

1. Gravitation; law of gravitation, gravity, weight.

2. Center of gravity.

3. The three states of equilibrium. Stability.

[Pg 109]

1. Why is a plumb-line useful in building houses?

2. What is the center of gravity of a body?

3. Explain the action of a rocking chair that has been tipped forward.

4. Is the stability of a box greater when empty or when filled with sand? Explain.

5. How can you start yourself swinging, in a swing, without touching the ground?

6. Is the center of gravity of the beam of a balance above, below, or at the point of a support? How did you find it out?

7. Why are some ink bottles cone shaped with thick bottoms?

8. Would an electric fan in motion on the rear of a light boat move it? Would it move the boat if revolving under water? Explain.

9. What turns a rotary lawn sprinkler?

10. Why, when you are standing erect against a wall and a coin is placed between your feet, can you not stoop and pick it up unless you shift your feet or fall over?

11. What would become of a ball dropped into a large hole bored through the center of the earth?

12. When an apple falls to the ground, does the earth rise to meet it?

13. How far from the earth does the force of gravity extend?

14. Why in walking up a flight of stairs does the body bend forward?

15. In walking down a steep hill why do people frequently bend backward?

16. Why is it so difficult for a child to learn to walk, while a kitten or a puppy has no such difficulty?

17. Explain why the use of a cane by old people makes it easier for them to walk?

94. Falling Bodies.—One of the earliest physical facts learned by a child is that a body unsupported falls toward the earth. When a child lets go of a toy, he soon learns to look for it on the floor. It is also of common observation that light objects, as feathers and paper, fall much slower than a stone. The information, therefore, that all bodies[Pg 110] actually fall at the same rate in a vacuum or when removed from the retarding influence of the air is received with surprise.

This fact may be shown by using what is called a coin and feather tube. On exhausting the air from this tube, the feather and coin within are seen to fall at the same rate. (See Fig. 78.) when air is again admitted, the feather flutters along behind.

95. Galileo's Experiment.—The fact that bodies of different weight tend to fall at the same rate was first experimentally shown by Galileo by dropping a 1-lb. and a 100-lb. ball from the top of the leaning tower of Pisa in Italy (represented in Fig. 79). Both starting at the same time struck the ground together. Galileo inferred from this that feathers and other light objects would fall at the same rate as iron or lead were it not for the resistance of the air. After the invention of the air pump this supposition was verified as just explained.

[Pg 111]

96. Acceleration Due to Gravity.—If a body falls freely, that is without meeting a resistance or a retarding influence, its motion will continually increase. The increase in motion is found to be constant or uniform during each second. This uniform increase in motion or in velocity of a falling body gives one of the best illustrations that we have of uniformly accelerated motion. (Art. 75.) On the other hand, a body thrown upward has uniformly retarded motion, that is, its acceleration is downward. The velocity acquired by a falling body in unit time is called its acceleration, or the acceleration due to gravity, and is equal to 32.16 ft. (980 cm.) per second, downward, each second of time. In one second, therefore, a falling body gains a velocity of 32.16 ft. (980 cm.) per second, downward. In two seconds it gains twice this, and so on.

In formulas, the acceleration of gravity is represented by "g" and the number of seconds by t, therefore the formula for finding the velocity, V,[F] of a falling body starting from rest is V = gt. In studying gravity (Art. 89) we learned that its force varies as one moves toward or away from the equator. (How?) In latitude 38° the acceleration of gravity is 980 cm. per second each second of time.

97. Experimental Study of Falling Bodies.—To study falling bodies experimentally by observing the fall of unobstructed bodies is a difficult matter. Many devices have been used to reduce the motion so that the action of a falling body may be observed within the limits of a laboratory or lecture room. The simplest of these, and in some respects the most satisfactory, was used by Galileo. It consists of an inclined plane which reduces the effective component of the force of gravity so that the motion of a body rolling down the plane may be observed[Pg 112] for several seconds. For illustrating this principle a steel piano wire has been selected as being the simplest and the most easily understood. This wire is stretched taut across a room by a turn-buckle so that its slope is about one in sixteen. (See Fig. 80.) Down this wire a weighted pulley is allowed to run and the distance it travels in 1, 2, 3, and 4 seconds is observed. From these observations we can compute the distance covered each second and the velocity at the end of each second.

In Fig. 63, if OG represents the weight of the body or the pull of gravity, then the line OR will represent the effective component along the wire, and OS the non-effective component against the wire. Since the ratio of the height of the plane to its length is as one to sixteen, then the motion along the wire in Fig. 80 will be one-sixteenth that of a falling body.

98. Summary of Results.—The following table gives the results that have been obtained with an apparatus arranged as shown above.

In this table, column 2 is the one which contains the results directly observed by the use of the apparatus. Columns, 3, 4, and 5 are computed from preceding columns.

[Pg 113]

Column 5 shows that the acceleration is uniform, or the same each second. Column 4 shows that the velocity increases with the number of seconds or that V = at. Column 3 shows that the increase in motion from 1 second to the next is just equal to the acceleration or 60 cm. This is represented by the following formula: s = 1/2 a(2t - 1).

The results of the second column, it may be seen, increase as 1:4:9:16, while the number of seconds vary as 1:2:3:4. That is, the total distance covered is proportional to the square of the number of seconds.

This fact expressed as a formula gives: S = 1/2at².

Substituting g, the symbol for the acceleration of gravity, for a in the above formulas, we have: (1) V = gt, (2) S = 1/2gt², (3) s = 1/2g(2t - 1).

99. Laws of Falling Bodies.—These formulas may be stated as follows for a body which falls from rest:

1. The velocity of a freely falling body at the end of any second is equal to 32.16 ft. per sec. or 980 cm. per second multiplied by the number of the second.

2. The distance passed through by a freely falling body during any number of seconds is equal to the square of the number of seconds multiplied by 16.08 ft. or 490 cm.

3. The distance passed through by a freely falling body during any second is equal to 16.08 feet or 490 cm. multiplied by one less than twice the number of the second.

[Pg 114]

1. Falling bodies.

2. Galileo's experiment.

3. Acceleration due to gravity.

4. Laws of falling bodies.

1. How far does a body fall during the first second? Account for the fact that this distance is numerically equal to half the acceleration.

2. (a) What is the velocity of a falling body at the end of the first second? (b) How far does it fall during the second second? (c) Account for the difference between these numbers.

3. What is the velocity of a falling body at the end of the fifth second?

4. How far does a body fall (a) in 5 seconds (b) in 6 seconds (c) during the sixth second?

5. (a) What is the difference between the average velocity during the sixth second and the velocity at the beginning of that second?

(b) Is this difference equal to that found in the second problem? Why?

6. A stone dropped from a cliff strikes the foot of it in 5 seconds. What is the height of the cliff?

7. Why is it that the increased weight of a body when taken to higher latitudes causes it to fall faster, while at the same place a heavy body falls no faster than a light one?

8. When a train is leaving a station its acceleration gradually decreases to zero, although the engine continues to pull. Explain.

9. Would you expect the motion of equally smooth and perfect spheres of different weight and material to be equally accelerated on the same inclined plane? Give reason for your answer. Try the experiment.

10. A body is thrown upward with the velocity of 64.32 ft. per sec. How many seconds will it rise? How far will it rise? How many seconds will it stay in the air before striking the ground?

11. 32.16 feet = how many centimeters?

12. The acceleration of a freely falling body is constant at any one place. What does this show about the pull which the earth exerts on the body?

[Pg 115]

100. The Simple Pendulum.—Any body suspended so as to swing freely to and fro is a pendulum, as in Fig. 81. A simple pendulum is defined as a single particle of matter suspended by a cord without weight. It is of course impossible to construct such a pendulum. A small metal ball suspended by a thread is approximately a simple pendulum. When allowed to swing its vibrations are made in equal times. This feature of the motion of a pendulum was first noticed by Galileo while watching the slow oscillations of a bronze chandelier suspended in the Cathedral in Pisa.

101. Definition of Terms. The center of suspension is the point about which the pendulum swings. A single vibration is one swing across the arc. A complete or double vibration is the swing across the arc and back again. The time required for a double vibration is called the period. The length of a simple pendulum is approximately the distance from the point of support to the center of the bob.

A seconds pendulum is one making a single vibration per second. Its length at sea-level, at New York is 99.31 cm. or 39.1 in., at the equator 39.01 in., at the poles 39.22 in.

A compound pendulum is one having an appreciable portion of its mass elsewhere than in the small compact body or sphere called a bob. The ordinary clock pendulum[Pg 116] or a meter stick suspended by one end are examples of compound pendulums.

The amplitude of a vibration is one-half the arc through which it swings, for example, the arc DC or the angle DAC in Fig. 81.

102. Laws of the Pendulum.—The following laws may be stated:

1. The period of a pendulum is not affected by its mass or the material of which the pendulum is made.

2. For small amplitudes, the period is not affected by the length of the arc through which it swings.

3. The period is directly proportional to the square root of the length. Expressed mathematically, t/t´ = √l/√l´.

103. Uses of the Pendulum.—The chief use of the pendulum is to regulate motion in clocks. The wheels are kept in motion by a spring or a weight and the regulation is effected by an escapement (Fig. 82). At each vibration of the pendulum one tooth of the wheel D slips past the prong at one end of the escapement C, at the same time giving a slight push to the escapement. This push transmitted to the pendulum keeps it in motion. In this way, the motion of the wheel work and the hands is controlled. Another use of the pendulum is in finding the acceleration of gravity, by using the formula, t = π√(l/g), in which t is the time in seconds of a single vibration and l the length of the pendulum. If, for example, the length of the seconds pendulum is 99.31 cm., then 1 = π√(99.31/g); squaring both sides of the equation, we have 1² = π²(99.31/g), or g = [Pg 117]π² × 99.31/1² = 980.1 cm. per sec., per sec. From this it follows that, since the force of gravity depends upon the distance from the center of the earth, the pendulum may be used to determine the elevation of a place above sea level and also the shape of the earth.

1. Simple pendulum.

2. Definitions of terms used.

3. Laws of the pendulum.

4. Uses of the pendulum.

1. What is the usual shape of the bob of a clock pendulum? Why is this shape used instead of a sphere?

2. Removing the bob from a clock pendulum has what effect on its motion? Also on the motion of the hands?

3. How does the expansion of the rod of a pendulum in summer and its contraction in winter affect the keeping of time by a clock? How can this be corrected?

4. Master clocks that control the time of a railway system have a cup of mercury for a bob. This automatically keeps the same rate of vibration through any changes of temperature. How?

5. How will the length of a seconds pendulum at Denver, 1 mile above sea-level, compare with one at New York? Why?

6. What is the period of a pendulum 9 in. long? Note. In problems involving the use of the third law, use the length of a seconds pendulum for l, and call its period 1.

7. A swing is 20 ft. high, find the time required for one swing across the arc.

8. A pendulum is 60 cm. long. What is its period?

9. If in a gymnasium a pupil takes 3 sec. to swing once across while hanging from a ring, how long a pendulum is formed?

10. A clock pendulum makes four vibrations a second, what is its length?

[Pg 118]

Force; definition, elements, how measured, units, dyne.

Graphic Representation; typical examples of finding a component, a resultant, or an equilibrant.

Motion; Laws of motion (3), inertia, curvilinear motion, centrifugal force, momentum, (M = mv), reaction, stress and strain.

Moment of Force; parallel forces, couple, effective and non-effective component.

Gravitation; law; gravity, center of; weight. Equilibrium 3 forms; stability, how increased.

Falling Bodies; velocity, acceleration, "g," Laws; V = gt, S = (1/2)gt² - s = (1/2)g(2t - 1).

Pendulum; simple, seconds, laws (3), t = π√(l/g).

[Pg 119]

104. Work.—"Whenever a force moves a body upon which it acts, it is said to do work upon that body." For example, if a man pushes a wheelbarrow along a path, he is doing work on it as long as the wheelbarrow moves, but if the wheelbarrow strikes a stone and the man continues to push and no motion results, from a scientific point of view he is then doing no work on it.

"Work signifies the overcoming of resistance," and unless the resistance is overcome no work is done. Lifting a weight is doing work on it, supporting a weight is not, although the latter may be nearly as tiresome as the former. Work as used in science is a technical term. Do not attach to it meanings which it has in every-day speech.

105. Measurement of Work.—Work is measured by the product of the force by the displacement caused in the direction of the force, that is W = fs. Therefore if a unit of force acts through a unit of space, a unit of work will be done. There are naturally several units of work depending upon the units of force and space employed.

English Work Unit.—If the force of one pound acts through the distance of one foot, a foot-pound of work is done. A foot-pound is defined as the work done when 1 lb. is lifted 1 ft. against the force of gravity.

Metric Work Unit.—If the force is one kilogram and the distance one meter, one kilogram-meter of work is done.

Absolute Work Unit.—If the force of one dyne acts through the distance of one centimeter a dyne-centimeter[Pg 120] of work is done. This usually is called an i. Other work units are sometimes used depending upon the force and distance units employed. One, the i, is equal to 10,000,000 ergs or 10⁷ ergs.

106. Energy.—In the various cases suggested in the paragraphs upon work, an agent, a man, an animal or a machine, was mentioned as putting forth an effort in order to do the work. It is also true that in order to perform work an agent must employ energy, or the energy of a body is its capacity for doing work. Where an agent does work upon a body, as in winding up a spring or in lifting a weight, the body upon which the work has been done may acquire energy by having work done upon it. That is, it may become able to do work itself upon some other body. For instance, a lifted weight in falling back to its first position may turn wheels, or drive a post into the ground against resistance; a coiled spring may run clock work, strike a blow, or close a door. Hence the energy, or the capacity for doing work, is often acquired by a body because work has first been done upon that body.

107. Potential Energy.—The wound up spring may do work because work has first been done upon it. The lifted weight may also do work because work has first been done in raising it to its elevated position since in falling it may grind an object to powder, lift another weight or do some other kind of work. The energy that a body possesses on account of its position or shape and a stress to which it is subjected is called potential energy. The potential energy of a body is measured by the work done in lifting it,[Pg 121] changing its shape, or by bringing about the conditions by which it can do work. Thus if a block of iron weighing 2000 lbs. is lifted 20 ft., it possesses 40,000 ft.-lbs. of potential energy. It is therefore able to do 40,000 ft.-lbs. of work in falling back to its first position. If the block just mentioned should fall from its elevated position upon a post, it could drive the post into the ground because its motion at the instant of striking enables it to do work. To compute potential energy you compute the work done upon the body. That is, P.E. = w × h or f × s.

108. Kinetic Energy.—The energy due to the motion of a body is called kinetic energy. The amount of kinetic energy in a body may be measured by the amount of work done to put it in motion. It is usually computed, however, by using its mass and velocity on striking. To illustrate, a 100-lb. ball is lifted 16 ft. The work done upon it, and hence its potential energy, is 1600 ft.-lbs. On falling to the ground again, this will be changed into kinetic energy, or there will be 1600 ft.-lbs. of kinetic energy on striking. It will be noted that since energy is measured by the work it can do, work units are always used in measuring energy. To compute the kinetic energy of a falling body by simply using its mass and velocity one proceeds as follows, in solving the above problem:

Important Topics

1. Work defined.

2. Work units, foot-pound, kilogram-meter, erg.

3. Energy defined.

4. Kinds of energy, potential and kinetic.

Problems

1. How much work will a 120-lb. boy do climbing a mountain 3000 ft. high? Should the vertical or slant height be used? Why?

2. In a mine 4000 kg. of coal are lifted 223 meters: how much work is done upon the coal? What is the kind and amount of energy possessed by the coal?

3. A pile driver weighs 450 lbs. It is lifted 16 ft. How much work has been done upon it? What kind and amount of energy will it have after falling 16 ft. to the pile?

4. A train weighing 400 tons is moving 30 miles per hour. Compute its kinetic energy. (Change its weight to pounds and velocity to feet per second.)

5. What would be the kinetic energy of the train in problem 4 if it were going 60 miles per hour? If it were going 90 miles per hour? How does doubling or trebling the speed of an object affect its kinetic energy? How does it affect its momentum?

6. What is the kinetic energy of a 1600-lb. cannon ball moving 2000 ft. per second?

7. Mention as many kinds of mechanical work as you can and show how each satisfies the definition of work.

8. A pile driver weighing 3000 lbs. is lifted 10 ft. How much work is done upon it?

[Pg 123]

9. If the pile driver in problem 8 is dropped upon the head of a pile which meets an average resistance of 30,000 lbs., how far will one blow drive it?

10. A 40 kg. stone is placed upon the top of a chimney 50 meters high. Compute the work done in kilogram-meters and foot-pounds.

(2) Power and Energy

109. Horse-power.—In computing work, no account is taken of the time required to accomplish it. But since the time needed to perform an undertaking is of much importance, the rate of work, or the power or activity of an agent is an important factor. Thus if one machine can do a piece of work in one-fifth the time required by another machine, it is said to have five times the power of the other. Therefore the power of a machine is the rate at which it can do work. James Watt (1736-1819), the inventor of the steam-engine, in expressing the power of his engine, used as a unit a horse-power. He considered that a horse could do 33,000 ft.-lbs. of work a minute. This is equal to 550 ft.-lbs. per second or 76.05 kg.-m. per second. This is too high a value but it has been used ever since his time. Steam engines usually have their power rated in horse-power. That is, locomotives produce from 500 to 1500 horse-power. Some stationary and marine engines develop as high as 25,000 horse-power. The power of an average horse is about 3/4 horse-power and of a man about 1/7 horse-power when working continuously for several hours.

110. The Watt.—In the metric system, the erg as a unit of work would give as a unit of power 1 erg per second. This amount is so small, however, that a larger unit is usually employed, the practical unit being 10,000,000 ergs a second, that is, one joule per second. (See Art. 105.) This practical unit is called a Watt after James Watt.[Pg 124] The power of dynamos is usually expressed in kilowatts, a kilowatt representing 1000 watts. Steam-engines in modern practice are often rated in kilowatts instead of horse-power. A horse-power is equivalent to 746 watts, or is nearly 3/4 of a kilowatt.

111. Energy. Its Transference and Transformation. We have considered energy as the capacity for doing work, and noted the two kinds, potential and kinetic, and the facility with which one may change into another. In fact, the transference of energy from one body to another, and its transformation from one form to another is one of the most common processes in nature. Take a pendulum in motion, at the end of a swing, its energy being entirely due to its elevated position is all potential; at the lowest point in its path its energy being entirely due to its motion is all kinetic. The change goes on automatically as long as the pendulum swings. A motor attached by a belt to a washing machine is started running. The energy of the motor is transferred by the belt to the washer where it is used in rubbing and moving the clothes.

The heat used in warming a house is usually obtained by burning coal or wood. Coal is believed to be formed from the remains of plants that grew in former geologic times. These plants grew through the help of the radiant energy of the sun. The following are transformations of energy that have occurred: The radiant energy of sunlight was transformed into the chemical energy of the plants. This remained as chemical energy while the plants were being converted into coal, was mined, brought to the stove or furnace and burned. The burning transformed the chemical energy into heat energy in which form we use it for warming rooms. Take the energy used in running a street car whose electrical energy comes from a waterfall. The energy of the car itself is mechanical.[Pg 125] Its motor, however, receives electrical energy and transforms it into mechanical. This electrical energy comes along a wire from a dynamo at the waterfall, where water-wheels and generators transform into electrical energy the mechanical energy of the falling water. The water obtained its energy of position by being evaporated by the heat of the radiant energy of the sun. The vapor rising into the air is condensed into clouds and rain, and falling on the mountain side, has, from its elevated position, potential energy. The order of transformation, therefore, is in this case, radiant, heat, mechanical, electrical, and mechanical. Can you trace the energy from the sun step by step to the energy you are using in reading this page?

112. Forms of Energy.—A steam-engine attached to a train of cars employs its energy in setting the cars in motion, i.e., in giving them kinetic energy and in overcoming resistance to motion. But what is the source of the energy of the engine? It is found in the coal which it carries in its tender. But of what kind? Surely not kinetic, as no motion is seen. It is therefore potential. What is the source of the energy of the coal? This question leads us back to the time of the formation of coal beds, when plants grew in the sunlight and stored up the energy of the sun's heat and light as chemical energy. The sun's light brings to the earth the energy of the sun, that central storehouse of energy, which has supplied nearly all the available energy upon the earth. Five forms of energy are known, viz., mechanical, heat, electrical, radiant, and chemical.

113. Energy Recognized by its Effects.—Like force, energy is invisible and we are aware of the forms only by the effects produced by it.

We recognize heat by warming, by expansion, by pressure.

[Pg 126]

We recognize light by warming, by its affecting vision.

We recognize electrical energy by its heat, light, motion, or magnetic effect. We recognize mechanical energy by the motion that it produces. We recognize chemical energy by knowing that the source of energy does not belong to any of the foregoing.

A boy or girl is able to do considerable work. They therefore possess energy. In what form does the energy of the body mainly occur? One can determine this for himself by applying questions to each form of energy in turn as in Art. 114.

114. Source of the Energy of the Human Body.—Is the energy of the human body mostly heat? No, since we are not very warm. Is it light or electrical? Evidently not since we are neither luminous nor electrical. Is it mechanical? No, since we have our energy even when at rest. Is it chemical? It must be since it is none of the others. Chemical energy is contained within the molecule.

It is a form of potential energy and it is believed to be due to the position of the atoms within the molecule. As a tightly coiled watch spring may have much energy within it, which is set free on allowing the spring to uncoil, so the chemical energy is released on starting the chemical reaction. Gunpowder and dynamite are examples of substances containing chemical energy. On exploding these, heat, light, and motion are produced. Gasoline, kerosene, and illuminating gas are purchased because of the potential energy they contain. This energy is set free by burning or exploding them.

The source of the energy of our bodies is of course the food we eat. The energy contained in the food is also chemical. Vegetables obtain their energy from the sunlight[Pg 127] (radiant energy). This is why plants will not grow in the dark. The available energy is mostly contained in the form of starch, sugar and oil. Digestion is employed principally to dissolve these substances so that the blood may absorb them and carry them to the tissues of the body where they are needed. The energy is set free by oxidation (burning), the oxygen needed for this being supplied by breathing. Breathing also removes the carbon dioxide, which results from the combustion. It is for its energy that our food is mostly required.

115. Conservation of Energy.—In the study of matter we learned that it is indestructible. Energy is also believed to be indestructible. This principle stated concisely teaches that despite the innumerable changes which energy undergoes the amount in the universe is unchangeable, and while energy may leave the earth and be lost as far as we are concerned, that it exists somewhere in some form. The principle which teaches this is called the "Conservation of Energy." The form into which energy is finally transformed is believed to be heat.

1. Power defined. Units. Horse-power. Watt.

2. Transference and transformations of energy.

3. Forms of energy; heat, electrical, mechanical, radiant, chemical.

4. Effects of the several forms of energy.

5. Energy of the human body.

6. Conservation of energy.

1. A boy weighing 110 lbs. ran up a stairs 10 ft. high, in 4 seconds. How much work was done? What was his rate of work (foot-pounds per second)? Express also in horse-power.

2. A locomotive drawing a train exerts a draw bar pull of 11,000 lbs. How much work does it do in moving 3 miles? What[Pg 128] is its rate of work if it moves 3 miles in 5 minutes? Express in horse-power.[G]

3. If 400 kg. are lifted 35 meters in 5 seconds what work is done? What is the rate of work? Express in horse-power, watts and kilowatts.

4. Trace the energy of a moving railway train back to its source in the sun.

5. Why does turning the propeller of a motor boat cause the boat to move?

6. Does it require more power to go up a flight of stairs in 5 seconds than in 10 seconds? Explain. Is more work done in one case than in the other? Why?

7. Can 1 man carrying bricks up to a certain elevation for 120 days do as much work as 120 men carrying up bricks for 1 day?

8. If the 1 man and 120 men of problem 7 do the same amount of work have they the same power? Explain.